J'.^X^^X 


University  of  California  •  Berkeley 


The  Theodore  P.  Hill  Collection 

0/ 

Early  American  Mathematics  Books 


ROBINSON'S    MATHEMATICAL    SERIES. 


THE 


RUDIMENTS 


OF 


WRITTEN   ARITHMETIC: 


CONTAINING 

SLATE  AND  BLACK-BOARD    EXERCISES  FOR  BEGINNERS, 
AND     DESIGNED     FOR    GRADED     SCHOOLS. 


EDITED  BY 

DANIEL  W.   FISH,   A.M. 


rVISON,  BLAKEMAN,  TAYLOR  &  CO., 

NEW  YORK  Aim  CHICAGO. 
1880. 


'    '     ROBINSON'S 

M  a  t  h  e  matie al    S  erie  s . 

Graded  to  the  wants  of  Primary,  Intermediate,  Grammar, 
Normal,  and  High  Schools,  Academies,  and  Colleges, 


«»t      — 

ProgressiTe  Table  Book. 
Progrressivo  Primary  Arithmetic. 
Progressive  Intellectual  Arithmetic. 
Railiments  of  Written  Arithmetic. 

JUNIOR-CLASS  ARITHMETIC,  Oral  and  Written.     NEW. 
Progressive  Practical  Arithmetic. 
Key  to  Practical  Arithmetic. 
Progressive  Higher  Arithmetic. 
Key  to  Higher  Arithmetic. 
Nen  Elementary  Algebra. 
iey  to  New  Elementary  Algebra. 
New  Tniveraity  Algebra. 
Key  to  New  University  Algebra, 
New  Geometry  and  Trigonometry.    In  one  vol. 
Geometry,  Plane  and  Solid.     In  separate  vol. 
Trigonometry,  Plane  acd  Spherical.    In  separate  vol. 
New  Analytical  Geometry  and  Conic  Sections. 
New  Surveying  and  Navigation. 
New  Differential  and  Integral  Calculas. 
University  Astronomy— Descriptive  and  Physical. 

Key  to  Geometry  and  Trigonometry,  Analytical  Geometry  and  Conic  Sec- 
tions, Surveying  and  Navigation, 

Entered,  according  to  Act  of  Congress,  in  the  year  1858,  and  again  in  the  year  1863,  by 

DANIEL    W.    FISH,    A.M., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United. States,  for  the  Northern 

District  of  New  York. 


Copyright^  1877,  by  Daniel  W.  Fish. 


PREFACE. 


JN  the  preparation  of  tliis  work,  a  special  object  has  been 
kept  in  view  by  the  author,  namely  :  to  furnish  a 
small  and  simple  class-book  for  beginners,  which  shall 
contain  no  more  of  theory  than  is  necessary  for  the  illus- 
tration and  apphcation  of  the  elementary  principles  of 
written  arithmetic,  appHed  to  numerous,  easy  and  prac- 
tical examples,  and  which  shall  be  introductory  to  a  full 
and  complete  treatise  on  this  subject. 

This  book  is  not  to  be  regarded  as  a  necessary  part  of 
the  Arithmetical  Series  by  the  same  author,  as  the  four 
books  already  composing  that  Series  are  belieyed  to  be 
properly  and  scientifically  graded,  and  eminently  adapted 
to  general  use  ;  but  this  work  has  been  prepared  to  meet 
a  limited  demand,  in  large  graded  schools,  and  in  the  pub- 
lic schools  of  New  York,  and  similar  cities,  where  a  large 
number  of  pupils  often  obtain  but  a  limited  knowledge  of 
arithmetic,  and  wish  to  commence  its  study  quite  young  ; 
and  it  is  also  designed  for  those  who  desire  a  larger  num- 
ber of  simple  and  easy  exercises  for  the  slate  and  black- 
board than  are  usually  found  in  a  complete  work  on  writ= 
ten  arithmetic,  so  that  the  beginner  may  acquire  facility, 
promptness,  and  accuracy  in  the  application  and  opera- 
tions of  the  fundamental  principles  of  this  science. 

(iii) 


IV  PKEFACE. 

The  principles,  definitions,  rules,  and  applications,  so  far 
as  developed  in  this  work,  coincide  with  the  other  books 
of  the  same  series.  Many  of  the  Contractions,  and  special 
applications  of  the  rules,  particularly  those  that  are  at  all 
difficult,  have  been  omitted,  and  also  the  treatment  of 
Denominate  Fractions,  and  Decimals,  all  of  which  are 
fully  and  practically  treated  in  the  Progressive  Practical, 
and  the  Higher  Arithmetic.  A  few  easy  and  practical 
applications  of  Cancellation,  Analysis,  Percentage,  and 
Simple  Interest  have  been  given,  and  a  very  large  number 
of  easy  examples. 


IMPROVED    EDITION. 

The  plates  of  this  book  having  become  much  worn,  and 
its  circulation  requiring  the  printing  of  large  editions, 
the  book  has  been  newly  electrotyped,  in  the  latest  and 
best  style  of  typography. 

The  text  of  the  previous  editions  has  been  preserved, 
unchanged,  page  for  page,  and  line  for  line,  as  nearly  as 
possible,  except  to  expunge  from  the  tables,  and  some 
other  portions  of  the  book,  such  matter  as  has  become 
entirely  obsolete  and  useless. 

No  changes  have  been  made,  that  will  at  all  interfere  in 
its  use,  with  any  of  the  previous  editions. 

April,  1877. 


CONTENTS. 


SIMPLE    NUMBERS. 

PAQH 

Defestttons 7 

Roman  Notation 8 

Arabic  Notation 9 

Laws  and  Rules  for  Notation  and  Numekation 16 

Addition 18 

Subtraction 29 

Multiplication 39 

Contractions 48 

DnisiON 54 

Contractions 68 

Problems  in  Simple  Integral  Numbers 72 

COMMON    FRACTIONS. 

Definitions,  Notation  and  Numeration 74 

Redctction  of  Fractions 78 

Addition  of  Fractions 83 

Subtraction  of  Fractions 86 

Multiplication  of  Fractions 88 

Division  of  Fractions 94 

DECIMALS, 

Notation  and  Numeration .102 

Reduction  op  Decimals 107 

Addition  of  Decimals 109 


y 


VI  CONTENTS. 

PAGE 
SUBTEACTION   OF  DECIMALS Ill 

Multiplication  of  Decimals 112 

Division  of  Decimals 114 

UNITED    STATES    MONEY. 

Reduction  of  United  States  Money 118 

Addition  of  U.  S.  Money  120 

Subtraction  of  U.  S.  Money 122 

Multiplication  of  U.  S.  Money 124 

Division  of  U.  S.  Money 125 

Bills 128 

COMPOUND    NUMBERS, 

WiliGHTs  AND  Measures 130 

Aliquot  Parts 145 

Reduction  Descending 146 

Reduction  Ascending 148 

Addition  of  Compound  Numbers 153 

Subtraction  of  Compound  Numbers 156 

Multiplication  of  Compound  Numbers 159 

Division  of  Compound  Nuivibers 162 

Cancellation 167 

Analysis .172 

Percentage 177 

Commission 179 

Profit  and  Loss 180 

Interest 181 

Promiscuous  Examples 186 

Metric  System 193 

Miscellaneous  Tables 197 


RUDIMEITS  OF  ARITHMETIC. 


DEFIXITIOE'S. 

1.  Quantity  is  any  thing  that  can  be  increased,  dimin- 
ished, or  measured;  as  distance,  space,  weight,  motion,  time. 

2.  A  Unit  is  one,  a  single  thing,  or  a  definite  quantity. 

3.  A  iN'uniber  is  a  unit,  or  a  collection  of  units. 

4.  An  Abstract  Number  is  a  number  used  without  ref- 
erence to  any  particular  thing  or  quantity  ;  as  3,  24,  756. 

5.  A  Concrete  Number  is  a  number  used  with  refer- 
ence to  some  particular  thing  or  quantity ;  as  21  hours, 
4  cents,  230  miles. 

6.  A  Simple  Number  is  either  an  abstract  number,  or 
a  concrete  number  of  bjit  one  denomination  ;  as  48, 
52  pounds,  36  days. 

7.  A  Compound  Number  is  a  concrete  number  ex- 
pressed in  two  or  more   denominations  ;   as  4  bushels 

3  pecks,  8  rods  4  yards  2  feet  3  inches. 

8.  An  Integral  Number,  or  Integ-er,  is  a  number 
which  expresses  whole  things  ;  as  5,  12  dollars,  17  men. 

9.  A  Fractional  Number,  or  Fraction,  is  a  number 
which  expresses  equal  parts  of  a  whole  thing  or  quantity ; 
as  "I-,  f  of  a  pound,  -^  of  a  bushel. 

10.  Like  Numbers  have  the  same  kind  of  unit,  or  ex- 
press the  same  kind  of  quantity.  Thus,  74  and  16  are 
like  numbers  ;    so  are  74  pounds  and  16  pounds  ;  also, 

4  weeks  3  days,  and  16  minutes  20  seconds,  both  being 
used  to  express  units  of  time. 


/ 


8  SIMPLE     NUMBEKS. 

11.  Unlike  lumbers  haye  different  kinds  of  units, 
or  are  used  to  express  different  kinds  of  quantity.  Thus, 
36  miles,  and  15  days  ;  5  hours  36  minutes,  and  7  bushels 
3  pecks. 

12.  Aritlimetic  is  the  Science  of  numbers,  and  the 
Art  of  computation. 

13.  The  Five  Funclaiiiental  Operations  of  Arith= 
metic  are,  Notation  and  Numeration,  Addition,  Subtrac- 
tion, Multiplication,  and  Division. 

NOTATION"   AND    NUMEEATIOK. 

4 

14.  Notation  is  a  method  of  writing  or  expressing 
numbers  by  characters  ;  and, 

15.  Numeration  is  a  method  of  reading  numbers 
expressed  by  characters. 

16.  Two  systems  of  Notation  are  in  general  use — the 
Roman  and  Arabic. 

THE    EOMAN    NOTATION 

IT.  Employs  seven  capital  letters  to  express  numbers, 
thus  : 
Letters.  I        V         X         L         0         D         M 

Values.  One,       Five,       Ten,      Fifty,  ^^2S?ed,hSyred,tl?ou' and. 

18.  The  Roman  notation  is  founded  upon  the  following 
principles  : 

1st.  Repeating  a  letter  repeats  its  value.  Thus,  II 
represents  two,  XX  twenty,  COO  three  hundred. 

2d.  If  a  letter  of  any  value  be  placed  after  one  of 
greater  value,  its  value  is  to  be  added  to  that  of  the  greater. 
Thus,  XI  represents  eleven,  LX  sixty,  DO  six  hundred. 

3d.  If  a  letter  of  any  value  be  placed  lefore  one  of  greater 


NOTATION     AN"D     KUMERATION". 


9 


value,  its  value  is  to  be  taken  from  that  of  the  greater. 
Thus,  IX  represents  nine,  XL  forty,  CD  four  hundred. 

4th.  If  a  letter  of  any  value  be  placed  between  two  let- 
ters, each  of  greater  value,  its  value  is  to  be  tahe7i  from  the 
sum  of  the  other  two.  Thus,  XIV  represents  fourteen, 
XXIX  twenty-nine,  XCIV  ninety-four. 

Table  of  Roma^^  Notation. 


I  is  One. 

XVIII  is  Eighteen. 

II  "  Two. 

XIX  "  Nineteen. 

Ill  "  Three. 

XX  "  Twenty. 

IV  "  Four. 

XXI  "  Twenty-one. 

V  "  Five. 

XXX  "  Thirty. 

VI  "  Six. 

XL  "  Forty. 

VII  "  Seven. 

L  "  Fifty. 

VIII  "  Eight. 

LX  "  Sixty. 

IX  "  Nine. 

LXX  "  Seventy. 

X  "  Ten. 

LXXX  "  Eighty. 

XI  "  Eleven. 

'*           XC  ''  Ninety. 

XII  "  Twelve. 

C  "  One  hundred. 

XIII  "  Thirteen. 

CC  "  Two  hundred. 

XIV  "  Fourteen. 

D  "  Five  hundred. 

XV  "  Fifteen. 

DC  "  Six  hundred. 

XVI  "  Sixteen. 

M  "  One  thousand. 

XVII  "  Seventeen. 

Express  the  following  numbers  by  the  Roman  notation; 

1.  Fourteen,  6.  Fifty-one. 

2.  Nineteen.  7.  Eighty-eight. 

3.  Twenty-four.  8.  Seventy-three. 

4.  Thirty-nine.  9.  Ninety-five. 

5.  Forty-six.  10.  One  hundred  one. 


THE    ARABIC    NOTATION 

19.  Employs  ten  characters  or  figures  to  express  numbers. 


2  tens, 

3  tens, 

4  tens. 

5  tens. 

or 

or 

or 

or 

Twenty, 

Thirty, 

Forty, 

Fifty, 

20; 

30; 

40; 

50; 

10  SIMPLE     KUM  BEES. 

Thus, 

FUfures.       0123456789 

Names,       cipher,  One,  Two,  Three,  Four,  Five,    Six,  Seven,  Eight,  Nine, 

20.  Tlie  first  character  is  called  naught,  because  it  has 
no  value  of  its  own.  The  other  nine  characters  are  called 
significant  figures,  because  each  has  a  value  of  its  own. 

21.  As  we  have  no  single  character  to  represent  ten,  we 
express  it  by  writing  the  unit,  1,  at  the  left  of  the  cipher, 
0,  thus,  10.     In  the  same  manner  we  represent 

6  tens,       7  tens,    8  tens,    9  tens. 

or  or  or  or 

Sixty,     Seventy,  Eighty,  Ninety. 

60;       70;       80;       90. 

22.  AYhen  a  number  is  expressed  by  two  figures,  the 
right  hand  figure  is  called  units,  and  the  left  hand  figure 
tens. 

We  express  the  numbers  between  10  and  20,  thus  : 

Eleven,Twelve,Thirteen,Fourteen,Fifteen,Sixteen,Seventeen,Eighteen,Nineteen 

11,      12,      13,      -14,      15,      IG,        17,        18,       19. 

In  like  manner  we  express  the  numbers  between  20  and 
30,  thus  :  21,  22,  23,  24,  25,  26,  27,  28,  29,  etc. 

The  greatest  number  that  can  be  expressed  by  two  figures 
is  99. 

23.  We  express  one  hundred  by  writing  the  unit,  1,  at 
the  left  hand  of  two  ciphers  ;  thus,  100.  In  like  manner 
we  write  two  hundred,  three  hundred,  etc.,  to  nine  hun- 
dred.    Thus  : 

One         Vwo       Three      Four       Five         Six        Seven      Eight       Nine 
kuncired,hundred,hundred,hundred,hundrecl,hundred,hundred,hundred,  hundred. 

100,     200,     300,     400,     500,     600,     700,     800,      900. 

24.  When  a  number  is  expressed  by  three  figures,  the 
right  hand  figure  is  called  units,  the  second  figure  tens, 
and  the  left  hand  figure  hundreds. 


NOTATIOIT     AND     KUMERATIOi^'.  11 

As  the  ciphers  have,  of  themselves,  no  vakie,  but  are 
always  used  to  denote  the  absence  of  value  in  the  places 
they  occupy,  we  express  tens  and  units  with  hundreds,  by 
writing,  in  place  of  the  ciphers,  the  numbers  representing 
the  tens  and  units.  To  express  one  hundred  fifty,  we 
Avrite  1  hundred,  5  tens,  and  0  units  ;  thus,  150.  To 
express  seven  hundred  ninety-two,  we  write  7  hundreds, 
9  tens,  and  2  units  ;  thus, 

TO 

W        H         t:^ 

7        9       2 

The  greatest  number  that  can  be  expressed  by  three 
figures  is  999. 

Express  the  following  numbers  by  figures  : — 

1.  Write  one  hundred  twenty-five. 

2.  Write  four  hundred  eighty-three. 

3.  Write  seven  hundred  sixteen. 

4.  Express  by  figures  nine  hundred. 

5.  Express  by  figures  two  hundred  ninety. 

6.  Write  eight  hundred  nine. 

7.  Write  five  hundred  five. 

8.  Write  five  hundred  fifty-seven. 

25.  We  express  one  thousand  by  writing  the  unit,  1,  at 
the  left  hand  of  three  ciphers  ;  thus,  1000.  In  the  same 
manner  we  write  two  thousand,  three  thousand,  etc.,  to 
nine  thousand  ;  thus. 

One  Two  Three  Four  Five  Six  Seven         Eight  Nine 

thousand,  thousand,  thousand,  thousand,  thousand,  thousand,  thousand,  thousand,  thousand. 

1000,  2000,  3000,  4000,  5000,  6000,  7000,  8000,  9000. 

26.  When  a  number  is  expressed  by  four  figures,  the 
places,  commencing  at  the  right  hand>  are  units,  tens, 
hundreds,  thousands. 


12  SIMPLE     NUMBERS. 

To  express  hundreds,  tens,  and  units  with  thousands,  we 
write  in  each  place  the  figure  indicating  the  number  we 
wish  to  express  in  that  place.  To  write  four  thousand 
two  hundred  sixty-nine,  we  write  4  in  the  place  of 
thousands,  2  in  the  place  of  hundreds,  6  in  the  place  of 
tens,  9  in  the  place  of  units  ;  thus, 


02 

• 

■TJ 

on 

a 

-^ 

CD 

§ 

m 

m 

o 

f^ 

g 

.rH 

rTj 

rs 

<B 

d 

Eh 

w 

Eh 

1::^ 

4      2      6      9 

The  greatest  number  that  can  be  expressed  by  four 
figures  is  9999. 

Express  the  following  numbers  by  figures  : — 

1.  One  thousand  two  hundred. 

2.  Five  thousand  one  hundred  sixty. 

3.  Three  thousand  seyen  hundred  forty-one. 

4.  Eight  thousand  fifty-six. 

5.  Two  thousand  ninety. 

6.  Seven  thousand  nine. 

7.  One  thousand  one. 

8.  Nine  thousand  four  hundred  twenty-seven. 

9.  Four  thousand  thirty-five. 

10.  One  thousand  nine  hundred  four. 
Read  the  following  numbers  : — 

11.  76;      128;      405;       910;       116;     8414;     1025. 

12.  2100;     5047;     7009;     4670;     3997;     1001. 

27.  JSText  to  thousands  comes  te7is  of  thousands,  and 
next  to  these  come  hundreds  of  thousands,  as  tens  and 
hundreds  come  in  their  order  after  units. 

Ten  thousand  is  expressed  by  removing  the  unit,  1,  one 
place  to  the  left  of  the  place  of  thousands,  or  by  writing  it 


NOTATION     AND     NUMERATION.  13 

at  the  left  hand  of  four  ciphers ;  thus,  10000  ;  and  one 
hundred  thousand  is  expressed  by  removing  the  unit,  1, 
still  one  place  farther  to  the  left,  or  by  writing  it  at  the 
left  hand  of  five  ciphers  ;  thus,  100000.  We  can  xpress 
thousands,  tens  of  thousands,  and  hundreds  of  thousands 
in  one  number,  in  the  same  manner  as  we  express  units, 
tens,  and  hundreds  in  one  number.  To  express  five 
hundred  twenty-one  thousand  eight  hundred  three,  we 
write  five  in  the  sixth  place,  counting  from  units,  2  in 
the  fifth  place,  1  in  the  fourth  place,  8  in  the  third  place, 
0  in  the  second  place,  (because  there  are  no  tens),  and  3 
in  the  place  of  units  ;  thus, 

O   m 

re  a         O^ 

*H     M  CD    cj 

fl  o       ^  g-       §        fl        §       .-g 


M  +^ 


xn 

n 

02 

a 

.  nJ 

c^ 

9 

CO 

f-t 

c 

• 

m 

a 
o 

H 

l-H 

H 

;::> 


5         2        1    .8      0      3 

The  greatest  number  that  can  be   expressed  by  jive 
figures  is  99999  ;  and  by  six  figures,  999999. 

Write  the  following  numbers  in  figures  : — 

1.  Twenty  thousand. 

2.  Forty-seven  thousand. 

3.  Eighteen  thousand  one  hundred. 

4.  Twelve  thousand  three  hundred  fifty. 

5.  Thirty-nine  thousand  five  hundred  twenty-two. 

6.  Fifteen  thousand  two  hundred  six. 

7.  Eleven  thousand  twenty  four. 

8.  Forty  thousand  ten. 

9.  Sixty  thousand  six  hundred. 

10.  Two  hundred  twenty  thousand. 

11.  One  hundred  fifty-six  thousand. 

12.  Eight  hundred  forty  thousand  three  hundred. 


14 


SIMPLE     KUMBERS. 


Read  the  following  numbers  : 

13.  5006;         12304;       96071;         5470;      203410. 

14.  36741;       400560;       13061;       49000;       100010. 
For  convenience  in  reading  large  numbers,  we  may  point 

tbem  oif,  by  commas,  into  periods  of  three  figures  each, 
counting  from  the  right  hand  or  unit  figure.  This  point- 
ing enables  us  to  read  the  hundreds,  tens,  and  units  in 
each  period  with  facility  as  seen  in  the  following 

ISTuMERATioiT  Table. 


Periods.     5th. 

Name    . 


4th. 


3d. 


50 

o 


m 

a 
o 


W 


% 


2d. 

m 

i 

o 


1st. 


5 


Orders 

OF 

IJl^"ITS. 


=!    5    fl 


m 


m 


O 

3g'3 


m 

^  o  ^ 

WEhD 


02 

3  a;  c! 


^  S  f5 


M 


en 


Number.       4  5,     3  7  0,    0  3  6,     4  0  8,     0  6  0 

^8.  Figures  occupying  different  places  in  a  number, 
as  units,  tens,  hundreds,  etc.,  are  said  to  express  different 
orders  of  units." 

29.  In  7iume7'ating,  or  expressing  numbers  verbally, 
the  various  orders  of  units  have  the  following  names  : 


Orders. 
1st  order  is  called 
2d   order 
3d   order 
4th  order 

5  th  order 

6  th  order  *^ 
7th  order 
8th  order 
9th  order 


a 


a 


a 


a 


a 


a 


a 


a 


(( 


(( 


(C 


a 


a 


a 


a 


Names. 

Units. 

Tens. 

Hundreds. 

Thousands. 

Tens  of  thousands. 

Hundreds  of  thousands. 

Millions. 

Tens  of  millions. 

Hundreds  of  millions. 


NOTATION^     AN"D     NUMERATION".  15 

"Write  and  read  the  following  numbers  : — 

1.  One  unit  of  the  third  order,  two  of  the  second,  five 
of  the  first.      A^is.  125  ;  read,  07ie  hundred  twenty-five. 

2.  Two  units  of  the  5th  order,  four  of  the  4th,  five  of 
the  2d,  six  of  the  1st. 

Ans.  24056  ;  read,  tiueiity-four  tJiousand fifty-six. 

3.  Seven  units  of  the  4th  order,  five  of  the  3rd,  three 
of  the  2d,  eight  of  the  1st. 

4.  Two  units  of  the  7th  order,  nine  of  the  6th,  four  of 
the  3d,  one  of  the  1st,  seven  of  the  2d. 

5.  Three  units  of  the  6th  order,  four  of  the  2d. 

6.  Nine  units  of  the  8th  order,  six  of  the  7th,  three  of 
the  5th,  seven  of  the  4th,  nine  of  the  1st. 

7.  Four  units  of  the  10th  order,  six  of  the  8th,  four  of 
the  7th,  two  of  the  6th,  one  of  the  3d,  five  of  the  2d. 

8.  Eight  units  of  the  12th  order,  four  of  the  11th,  six 
of  the  10th,  nine  of  the  7th,  three  of  the  6th,  five  of  the 
5th,  two  of  the  3d,  eight  of  the  1st. 

SO.  Since  the  number  expressed  by  any  figure  depends 
upon  the  place  it  occupies,  it  follows  that  figures  have  two 
values.  Simple  and  Local. 

31.  The  Simple  Value  of  a  figure  is  its  value  when 
taken  alone ;  thus,  4,  7,  2. 

32.  The  Local  Value  of  a  figure  is  its  value  when 
used  with  another  figure  or  figures  in  the  same  number. 
Thus,  in  325,  the  local  value  of  the  3  is  300,  of  the  2  is  20, 
and  of  the  5  is  5  units. 

When  a  figure  occupies  units'  place,  its  simple  and  local  valuee  are  the 
same. 

33.  The  leading  principles  upon  which  the  Arabic 
notation  is  founded  are  embraced  in  the  following 


16  SIMPLE     KUMBEKS. 


Gei^ekal  Laws. 


I.  All  numbers  are  expressed  dy  applying  the  ten  figures 
.  to  the  different  orders  of  units. 

II.  The  different  orders  of  units  increase  from  right  to 
left,  and  decrease  from  left  to  right,  in  a  tenfold  ratio. 

III.  Every  7'emoval  of  a  figure  one  place  to  the  left,  in- 
creases its  local  value  tenfold  ;  and  every  removal  of  a  fig- 
ure one  place  to  the  right,  diminishes  its  local  value  tenfold. 

From  this  analysis  of  the  principles  of  IN'otation  and 
Numeration,  we  derive  the  following  rules  : — 

EuLE  EOE  Notation?". 

I.  Begi7ining  at  the  left  hand,  ivrite  the  figures  belonging 
to  the  highest  period. 

II.  Write  the  hundreds,  tens,  and  units,  of  each  succes- 
sive period  in  their  order,  placing  a  cipher  wherever  an 
'Order  of  units  is  omitted. 

Rule  for  JSTumeratioi^. 

I.  Separate  the  numher  into  periods  of  three  figures  each, 
commencing  at  the  right  hand. 

II.  Beginning  at  the  left  hand,  read  each  period  sepa- 
rately, and  give  the  name  to  each  period,  except  the  last,  or 
period  of  units. 

34.  Until  the  pupil  can  write  numbers  readily,  it  may 
be  well  for  him  to  write  several  periods  of  ciphers,  point 
them  off,  over  each  period  write  its  name,  thus. 

Trillions,  Billions,  Millions,        Thousands.         Units. 

000,         000,         000,         000,         000. 


NOTATION     AIs'D     KUMERATIOK.  17 

and  then  write  the  given  numbers  underneath,  in  their 
appropriate  places. 

Exercises  lis  NoTATioi^  axd  Numerations". 

Express  the  following  numbers  by  figures  : — 

1.  Four  hundred  thirty-six. 

2.  Seven  thousand  one  hundred  sixty-four. 

3.  Twenty-six  thousand  twenty-six. 

4.  Fourteen  thousand  two  hundred  eighty. 

5.  One  hundred  seventy-six  thousand. 

6.  Four  hundred  fifty  thousand  thirty-nine. 

7.  Ninetj'-five  million. 

8.  Four  hundred  eighty-three  million  eight  hundred 
sixteen  thousand  one  hundred  forty-nine. 

9.  Xine  hundred  thousand  ninety. 

10.  Ten  milhon  ten  thousand  ten  hundred  ten. 

Point  off,  numerate,  and  read  the  following  numbers : — 


11.  8240. 

12.  400900. 

13.  308. 

14.  60720. 


15.  mill. 

16.  57468139. 

17.  5623. 

18.  11111111. 


19.  370005. 

20.  9400706342. 

21.  38429526. 

22.  11111111111 


23.  AVrite  seven  million  thirty-six. 

24.  Write  five  hundred  sixty-three  thousand  four. 

25.  Write  one  million  ninety-six  thousand. 

26.  A  certain  number  contains  3  units  of  the  seventh 
order,  6  of  the  fifth,  4  of  the  fourth,  1  of  the  third,  5  of 
the  second,  and  2  of  the  first ;  what  is  the  number  ? 

27.  What  orders  of  units  are  contained  in  the  number 
290648  ? 


18 


SIMPLE     LUMBERS. 


ADDITIOjST, 

35.  Addition  is  the  process  of  uniting  several  num- 
bers of  the  same  kind  into  one  equivalent  number. 

36.  The  Sum  or  Amount  is  the  result  obtained. 

Addition  Table. 


2  and    1  are 

3 

3  and    1  are 

4 

4  and 

1  are 

5 

5  and 

1  are    6 

2  and    2  are 

4 

3  and    2  are 

5 

4  and 

2  are 

6 

5  and 

2  are    7 

2  and    3  are 

5 

3  and    3  are 

6 

4  and 

3  are 

7 

5  and 

3  are    8 

2  and    4  are 

6 

3  and    4  are 

7 

4  and 

4  are 

8 

5  and 

4  are    9 

2  and    5  are 

7 

3  and    5  are 

8 

4  and 

5  are 

9 

5  and 

5  are  10 

2  and    G  are 

8 

3  and    6  are 

9 

4  and 

6  are 

10 

5  and 

0  are  11 

2  and    7  are 

9 

3  and    7  are 

10 

4  and 

7  are 

11 

5  and 

7  are  12 

2  and    8  are 

10 

3  and    8  are 

11 

4  and 

Bare 

12 

5  and 

8  are  13    1 

2  and    9  are 

11 

3  and    9  are 

12 

4  and 

9  are 

13 

5  and 

9  are  14    1 

2  and  10  are 

12 

3  and  10  are 

13 

4  and 

10  are 

14 

5  and  10  are  15 

2  and  11  are 

13 

3  and  11  are 

14 

4  and  11  are 

15 

5  and  11  are  16 

2  and  12  are 

14 

3  and  12  are 

15 

4  and 

12  are 

16 

5  and  12  are  17 

6  and    1  are 

7 

7  and    1  are 

8 

8  and 

1  are 

0 

9  and 

1  are  10 

6  and    2  are 

8 

7  and    2  are 

9 

8  and 

2  are 

10 

0  and 

2  are  11 

6  and    3  are 

9 

7  and    3  are 

10 

8  and 

3  are 

11 

9  and 

3  arc  12 

6  and    4  are 

10 

7  and    4  are 

11 

8  and 

4  are 

12 

9  and 

4  are  13 

6  and    5  are 

11 

7  and    5  are 

12 

8  and 

5  are 

13 

9  and 

5  are  14 

6  and    6  are 

12 

7  and    6  are 

13 

8  and 

6  are 

14 

9  and 

6  are  15 

G  and    7  are 

13 

7  and    7  are 

14 

8  and 

7  are 

15 

9  and 

7  are  16 

C  and    8  are 

14 

7  and    8  are 

15 

8  and 

8  are 

16 

9  and 

8  are  17 

6  and    9  are 

15 

7  and    9  are 

16 

8  and 

9  are 

17 

9  and 

9  are  18 

fi  and  10  are 

16 

7  and  10  are 

17 

8  and 

10  are 

IS 

9  and  10  are  19 

G  and  11  are 

17 

7  and  11  are 

18 

8  and 

11  are 

19 

9  and 

11  are  20 

G  and  12  are 

18 

7  and  12  are 

19 

8  and  12  are 

20 

9  and 

12  are  21 

10  and    1  are 

11 

11  and    1  are 

12 

12  and 

1  are 

13 

13  and 

1  are  14 

10  and    2  are 

12 

11  and    2  are 

13 

12  and 

2  are 

14 

13  and 

2  are  15 

10  and    3  are 

13 

11  and    3  are 

14 

12  and 

3  are 

15 

13  and 

3  are  16 

10  and    4  are 

14 

11  and    4  are 

15 

12  and 

4  are 

16 

13  and 

4  are  17 

10  and    5  are 

15 

11  and    5  are 

16 

12  and 

5  are 

17 

13  and 

5  are  18 

10  and    G  are 

16 

11  and    6  are 

17 

12  and 

6  are 

18 

13  and 

6  are  19 

10  and    7  are 

17 

1 1  and    7  are 

18 

12  and 

7  are 

19 

13  and 

7  are  20 

10  and    8  are 

18 

11  and    8  are 

19 

12  and 

8  are 

20 

13  and 

8  are  21 

10  and    9  are 

19 

11  and    9  are 

20 

12  and 

9  are 

21 

13  and 

9  are  22 

10  and  10  are 

20 

11  afid  10  are 

21 

12  and  10  are 

22 

13  and  10  are  23 

10  and  11  are 

21 

11  and  11  are 

22 

12  and 

11  are 

23 

13  and  11  are  24 

10  and  12  are 

22 

11  and  12  are 

23 

12  and  12  are 

24 

13  and 

12  are  25 

addition".  19 

Me:n'tal  Exekcises. 

1.  A  farmer  paid  6  dollars  for  a  straw-cutter,  and  9  dol- 
lars for  a  plow  ;  what  did  he  pay  for  both  ? 

Analysis.  He  paid  the  sum  of  6  dollars  and  9  dollars,  wMch  is 
15  dollars. 

2.  John  gave  4  apples  to  James,  8  to  Henry,  and  9  to 
Asa  ;  how  many  did  he  give  to  all  ? 

3.  I  gave  7  dollars  for  a  barrel  of  flour,  9  dollars  for  a 
hundred  weight  of  sugar,  and  6  dollars  for  a  tub  of  butter; 
what  did  I  give  for  the  whole  ? 

4.  I  have  two  pear  trees  ;  one  tree  produced  12  bushels 
of  pears,  and  the  other  11  bushels  ;  how  many  bushels  did 
both  produce  ? 

5.  A  man  bought  4  €ords  of  wood  for  12  dollars,  and 
7  bushels  of  corn  for  5  dollars ;  what  did  he  pa}^  for 
both? 

6.  James  gave  11  cents  for  a  slate,  and  had  8  cents 
left ;  how  many  cents  had  he  at  first  ? 

7.  A  lady  paid  5  dollars  for  a  bonnet,  10  dollars  for  a 
shawl,  and  had  7  dollars  left ;  how  much  money  had  she 
at  first  ? 

8.  In  a  shop  are  8  men,  9  boys,  and  6  girls,  at  work ; 
how  many  persons  are  at  work  in  the  shop  ? 

9.  Rollin  bought  a  quire  of  paper  for  1 2  cents,  a  slate 
for  13  cents,  and  gave  10  cents  to  a  beggar ;  how  much 
money  did  he  pay  out  in  all  ? 

10.  A  man  bought  4  bushels  of  wheat  for  7  dollars, 
18  bushels  of  corn  for  11  dollars,  and  2  cords  of  wood  for 
5  dollars  ;  what  did  he  pay  for  the  whole  ? 

11.  A  farmer  has  6  cows  in  one  yard,  9  in  another,  and 
as  many  in  the  third  yard  as  in  both  the  others  ;  how 
many  cows  has  he  ? 


20 


SIMPLE     ]S"UMBERS. 


Promiscuous  Additiok  Table. 


2  and  5  are  how  many  ? 
6  and  2  are  how  many  ? 
2  and  4  are  how  many  ? 

8  and  9  are  how  many  ? 

9  and  4  are  how  many  ? 
4  and  7  are  how  many  ? 
8  and  6  are  how  many  ? 

6  and  3  are  how  many  ? 

7  and  2  are  how  many  ? 


3  and  9  are  how  many? 

4  and  5  are  how  many  ? 
9  and  8  are  how  many  ? 
8  and  5  are  how  many  ? 

4  and  9  are  how  many  ? 

5  and  4  are  how  many  ? 
2  and  7  are  how  many  ? 
7  and  5  are  how  many? 
5  and  2  are  how  many  ? 


6  and  9  are  how  many  ? 

7  and  7  are  how  many  ? 

3  and  4  are  how  many  ? 

8  and  7  are  how  many  ? 

4  and  8  are  how  many  ? 

9  and  2  are  how  many  ? 

5  and  3  are  how  many  ? 

6  and  6  are  liow  many  ? 

7  and  4  are  how  many  ? 


7  and  9  are  how  many  ? 

6  and  5  are  how  many? 

3  and  6  are  how  many  ? 

4  and  4  are  how  many  ? 

7  and  8  are  how  many  ? 
9  and  3  are  how  maiiy  ? 

5  and  4  are  how  many  ? 
3  and  8  are  how  many? 
5  and  6  are  how  many  ? 


5  -and  8  are  how  many  ? 
3  and  7  are  how  many  ? 

6  and  4  are  how  many  ? 

7  and  6  are  how  many? 
6  and  8  are  how  many  ? 
9  and  5  are  how  many  ? 

8  and  3  are  how  many  ? 

9  and  6  are  how  many? 
5  and  7  are  how  many  ? 


4  and  6  are  how  many  ? 

7  and  3  are  how  many? 
2  and  8  are  how  many  ? 

5  and  9  are  how  many  ? 

8  and  8  are  how  many  ? 

6  and  7  are  how  many? 
5  and  5  are  how  many  ? 

9  and  7  are  how  many  ? 
9  and  9  are  how  many  ? 


37.  The  Sig-n  of  Addition  is  the  perpendicular  cross, 
+ ,  called  plus.  It  shows  that  the  numbers  connected  by 
it  are  to  be  added  ;  as  3  -f-  5  +  7^  read  3  'plus  5  plus  7. 

38.  The  Sign  of  Equality  is  two  short,  parallel,  hori- 
zontal lines,  =.  It  shows  that  the  numbers,  or  combina- 
tion of  numbers,  connected  by  it  are  equal ;  as  4  +  8  == 
9  +  3,  read  the  sum  of  4  plus  8  is  equal  to  the  sum  of  9 
plus  3. 


ADDITI0:5T.  21 


Case  I. 


39.    Wben   the   amount   of  each   column    is  less 
than  10. 

1.  A  droTer  bought  three  flocks  of  sheep.     The  first 

contained  232,  the  second  422,  and  the  third  245  ;  how 

many  sheep  did  he  buy  in  all  ? 

OPEKATiON.  Analysis.     Arrange  tlie  numbers  so  that 

®      •  units  of  like  order  shall  stand  in  tlie  same 

eg 'I  column.     Then  add   the  columns  separately, 

^  for  convenience    commencing    at    the    right 


232  lor   convenience    commeuciug 

hand,  and  write  each  result  under  the  column 
added.     Thus,  we  have  5  and  2  and  2  are  9, 
the  sum  of  the  units  ;    4  and  2  and  3  are  9, 
Amount,  899  the  sum  of  the  tens ;  2  and  4  and  2  are  8,  the 

sum  of    the  hundreds.  —  Hence,    the    entire 


422 
245 


amount  is  8  hundreds  9  tens  and  9  units,  or  899. 

Examples  foe  Peactice. 


(2.) 

(3.) 

(4-) 

(5.) 

403 

164 

510 

•   234 

271 

.  321 

176 

324 

124 

510 

203 

140 

A)is.   798 

(6.) 

(7.) 

(8.) 

(9.) 

1234 

2041 

3102 

4100 

2405 

3216 

2253 

1523 

5140 

1500 

4014  , 

2041 

Ajis.   8779 

10.  What  is  the  sum  of  421,  305  and  5162  ? 

11.  What  is  the  sum  of  3121,  436  and  2002  ? 


22  SIMPLE     NUMBEKS. 

Case  II. 

40.  "When  the  amount  of  any  column  equals  or 
exceeds  10. 

I.  A  merchant  pays  397  dollars  for  freights,  476  dollars 
for  a  clerk,  and  873  for  rent  of  a  store ;  what  is  the  amount 
of  his  expenses  ? 

OPERATION.  Analysis.    Arrange  the  numbers  so  that  units 

397  of  li^e  order  shall  stand  in  the  same  column. 

AtvQ  Then  add  the  first,  or  right  hand  column,  and 

g^„  the  sum  is  16  units,  or  1  ten  and  6  units  ;  writing 

.  the  6  units  under  the  column  of  units,  add  the 

1746  1  ten   to  the   column  of  tens,  and  the  sum  is 

24  tens,  or  2  hundreds  and  4  tens  ;  writing  the 
4  tens  under  the  column  of  tens,  add  the  2  hundreds  to  the  column 
of  hundreds,  and  the  sum  is  17  hundreds,  or  1  thousand  and  7  hun- 
dreds ;  writing  the  7  hundreds  under  the  column  of  hundreds,  and 
the  1  in  thousands'  place,  we  have  the  entire  sum,  174G. 

1.  lu  adding,  learn  to  prononnce  the  partial  results  without  narainj?  the 
figures  separately.    Thus,  in  the  operation  given  for  illustration,  say  '6,  9, 16 ;  8, 

'15,24;  10,  14,  17. 

2.  When  the  sum  of  any  column  is  greater  than  9,  the  process  of  adding  the 
tens  to  the  next  column  is  called  carrying. 

41.  From  the  preceding  examples  and  illustrations  we 
deduce  the  following 

EuLE.  I.  Write  the  numbers  to  ie  added  so  tliat  tlie 
units  of  the  same  order  shall  stand  in  the  same  column  ; 
that  is,  units  under  units,  tens  under  tens,  etc. 

II.  Commencing  at  units,  add  each  column  separately, 
and  write  the  sum  underneath,  if  it  he  less  than  ten. 

III.  If  the  sum  of  any  column  he  ten  or  more  than  ten, 
write  the  unit  figure  only,  and  add  the  ten  or  tens  to  thd 
next  column, 

IV.  Write  the  entire  sum  of  the  last  column. 


,.x. . 

ADDITION.  23 

Peoof.  Begin  with  the  right  hand  or  unit  column, 
and  add  the  ligures  in  each  cokimn  in  an  opposite  direc- 
tion from  that  in  which  tijey  were  first  added  ;  if  the  two 
results  agree,  the  work  is  supposed  to  be  right. 

Examples  for  Practice. 


(1-)   ' 

(2.) 

(3.) 

(4.) 

(5.) 

Inches. 

Feet. 

Pounds. 

Yards. 

Miles. 

142 

325 

75 

407 

1270 

325 

46 

276 

90 

342 

476 

C74 

508 

2584 

79 

943 

1045 

859 

3087 

1091 

(6.) 

(7.) 

(8.) 

(9.) 

(10.) 

842 

376 

426 

713 

4701 

390 

407 

397 

86 

374 

472 

802 

450 

345 

83 

205 

94 

294 

00 

19 

11.  What  is  the  sum  of  912  +  342  +  187+46  ? 

Ans,  1487. 
1^.  What  is  the  sum  of  214  +  425  +  90  +  37  ? 

13.  What  is  the  sum  of  50  feet,  450  feet,  and  080  feet  ? 

Ans.  1180  feet. 

14.  What  is  the  sum  of  1942  dollars,  and  085  dollars  ? 

15.  A  man  paid  375  dollars  for  a  span  of  horses, 
100  dollars  for  a  carriage,  and  87  dollars  for  a  harness  ; 
what  did  he  pay  for  all  ?  Am^.  622  dollars. 

16.  A  man  traveled  470  miles  by  railroad,  390  miles  by 
steamboat,  and  120  miles  by  stage  ;  how  many  miles  in 
all,  did  he  travel?  Ans.  980  miles. 

17.  A  carpenter  built  a  house  for  2404  dollars,  a  barn 
for  496  dollars,  and  out-houses  for  309  dollars  ;  what  did 
he  receive  for  building  all  ? 


24 


SIMPLE     KUMBEKS. 


18.  A  merchant  bought  at  public  auction  520  yards  of 
broadcloth,  386  yards  of  muslin,  92  yards  of  flannel,  and 
156  yards  of  silk  ;  how  many  yards  in  all  ? 

19.  A  father  divided  his  estate  among  his  four  sons,  giv- 
ing each  2087  dollars  ;  what  was  the  amount  of  his  estate  ? 

20.  Three  persons  deposited  money  in  a  bank  ;. the  first 
4780  dollars,  the  second  3042  dollars,  and  the  third 
407  dollars  :  how  much  did  they  all  deposit  ? 

21.  Five  men  engage  in  business  as  partners,  and  each 
puts  in  2375  dollars  ;  what  is  the  whole  amount  ol  capital 
invested?  A'ns.  11875  dollars. 


(22.) 

(23.) 

(24.) 

(25.) 

765 

347 

630 

4603 

381 

192 

815 

7106 

976 

763 

456 

972 

315 

410 

307 

385 

169 

507 

960 

64 

Ans.  2006 

(26.) 

(27.) 

(28.) 

767346 

374205 

4076315 

432761 

108497 

5632870 

386109 

643024 

8219634 

508703 

879638 

3827692 

i?25. 2094979 

29.  3720  +  647  +  190  +  82  "how  many? 

Ans.  4639. 

30.  962  +  2161  +  500  +  75  =  how  many  ? 

Ans.  3698. 

31.  4170  +  1009  +  642  +  120  +  18  =  how  many? 

32.  3000  +  47602  +  805  + 1266  +  76  =  how  many  ? 

33.  69  +  4030  +  349  + 1384  +  72  +  400  =  how  mauy  ? 


ADDITION".  25 

34.  What  is  tlie  sum  of  two  thousand  eight  hundred 
fifty-six,;  twelve  thousand  eighty-four,  seven  hundred 
forty-two,  and  sixty-nine  ?  Ans.  15751. 

35.  AVhat  is  the  amount  of  twenty  thousand  five  hun- 
dred ten,  six  thousand  nine  hundi'ed  forty-four,  and  three 
thousand  two  hundred  ?  Ans.  30654. 

36.  What  is  the  sum  of  forty-seven  thousand  fifty,  nine 
thousand  one  hundred  six,  fourteen  hundred  ninety-two, 
and  five  hundred  twelve  ?  Ans.  58160. 

37.  What  is  the  sum  of  one  hundred  forty  thousand 
three  hundred  thirty-four,  seventy-nine  thousand  six  hun- 
dred five,  twenty-five  hundred  twenty-five,  and  three 
thousand  sixty-iiine  ?  Ans.  225533. 

38.  What  is  the  amount  of  five  hundred  thousand  five 
hundred  five,  eighty-four  thousand  two  hundred,  fifteen 
thousand  six  hundred  twenty,  and  seventeen  hundred 
seventeen?  Ans.  602042. 

39.  How  many  men  in  an  army  consisting  of  26840 
infantry,  6370  cavalry,  3250  dragoons,  750  artillery,  and 
320  miners  ?  A7is.  37530. 

40.  A  merchant  deposited  125  dollars  in  bank  on  Mon- 
day, 91  on  Tuesday,  164  on  Wednesday,  200  on  Thursday, 
196  on  Friday,  and  73  on  Saturday  ;  bow  much  did  he 
deposit  during  the  week  ? 

41.  By  solliDi;-  a  farm  for  3586  dollars,  684  dollars  were 
lost ;  what  did  the  farm  cost  ?  ii  U  I  tj 

42.  If  I  were  horn  in  1840,  when  will  ]f  be  63  years  old?. 

43.  A  man  willed  his  estate  to  his  wife,  two  sons  and 
three  daughters  ;  to  his  daughters  he  gave  1565  dollars 
apiece,  to  his  sons  3560  dollars  each,  and  to  his  wife  4720 
dollars  ;  what  was  his  estate  worth?    Ans.  16535  dollars. 

44.  A  man  engaging  in  trade,  gained  450  dollars  the  first 


26  SIMPLE     NUM  BEES. 

year,  684  dollars  the  second,  and  as  much  the  third  as  he 
gained  during  the  first  and  second  ;  wliat  was  his  whole 
gain  ?  Ans,  2268  dollars. 

V.^  45.  I  bought  three  village  lots  for  12570  dollars,  and 
sold  them  so  as  to  gain  745  dollars  on  each  lot  ;  for  how 
much  did  I  sell  them  ?  Ans.  14805  dollars. 

46.  A  has  3240  dollars,  B  has  5672  dollars,  and  C  has 
1000  more  than  A  and  B  together  :  how  many  dollars 
have  all?  A71S.  18824  dollars. 

47.  A  man  was  32  years  old  when  his  son  was  born;  how 
old  will  he  be  when  his  son  is  36  years  old  ?  Ans.  68  years. 

48.  The  Old  Testament  contains  39  books,  929  chapters, 
23214  vor:-,.,  592439  words,  and  2728100  letters;  the 
New  Testament  contains  37  books,  269  chapters,  7959 
verses,  181153  words,  and  838380  letters  ;  what  is  the 
total  number  of  each  in  the  Bible  ? 

Ans.  76  books,  1198  chapters,  31173  verses,  773592 
words,  and  3566480  letters. 

49.  The  number  of  immigrants  landed  in  New  York  in 
1858  was  78589,  in  1859,  79322,  and  in  1860,  103621; 
what  ^v3iB  the  lotai  number  landed  in  the  three  years  Y 

Ans.  261532. 

50.  In  IBfiO,  the  population  of  New  York  was  814277, 
of  Philadelphia  568034,  of  Bosl  ^t  _7/902,  of  New  Orleans 
170766,  of  St.  Louis  162179.  .  iiicago  109429,  and  of 
Cincinnati '  160000  ;  what  was  the  total  population  of 
these  cities?  Ans.  2162587. 

51.  In  the  year  1856,  the  United  States  exported  molasses 
to  the  value  of  154630  dollars  ;  in  1857.  108003  dollars  ; 
in  1858, 115893  dollars;  what  was  the  value  of  the  molasses 
exported  in  those  three  years  ?       Ans.  378526  dollars. 

52.  During  the  same  years, respectively,  the  United  States 


ADDITIOI^r.  27 

exported  tobacco  to  tlie  value  of  1829207  dollars,  1458553 
dollars, and  2410224  dollars;  what  was  the  total  value  of  the 
tobacco  exported  in  those  years  ?    Ans.  5G97984  dollars. 

53.  How  many  miles  from  the  southern  extremity  of 
Lake  Michigan  to  the  Gulf  of  St.  Lawrence,  passing* 
through  Lake  Michigan,  330  miles;  Lake  Huron,  2G0 
miles;  River  St.  Clair,  24  miles;  Lake  St.  Clair,  20  miles; 
Detroit  River,  23  miles;  Lake  Erie,  260  miles;  Niagara 
River,  34  miles;  Lake  Ontario,  180  miles;  and  the  River 
St.  Lawrence,  750  miles  ?  Ans.  1881  miles. 

54.  At  the  commencement  of  the  year  1858  there  w^ere 
in  operation  in  the  Kew  England  States,  3751  miles  of  rail- 
road; in  New  York,  2590  miles;  in  Pennsylvania,  2546; 
in  Ohio,  2946;  in  Virginia,  1233;  in  Illinois,  2678;  and 
in  Georgia,  1233 ;  what  was  the  aggregate  number  of  miles 
in  operation  in  all  these  States  ?       .  Ans.  16977. 

55.  The  number  of  pieces  of  silver  coin  made  at  the 
United  States  Mint  at  Philadelphia  in  the  year  1858,  were 
as  follows:  4628000  half  dollars,  10600000  quarter  dollars, 
690000  dimes,  4000000  half  dimes,  and  1266000  three- 
cent  pieces;  what  was  the  total  number  of  pieces  coined  ? 

tAjis.  21184000. 


(56.) 

(57.) 

(58.) 

(59.) 

344 

843 

1186 

81988 

388 

738 

513 

380167 

613 

237 

740 

108424 

803 

218 

1820 

193686 

825 

347 

955 

144225 

412 

288 

736 

112558 

322 

483 

810 

107481 

886 

753 

511 

176826 

620 

834 

1179 

145851 

5213 

^yi' 

8450 

1451206 

28 


SIMPLE    :NrUMBEIlS 


*^ 


(60.) 

■  (61.) 

(62.) 

(63.) 

35938 

47197 

12380 

456568 

49172 

63956 

98795 

754712 

56546 

85678 

23442 

567346 

82564 

35495 

87639 

543678 

69789 

16457 

91758 

342766 

47321 

94667 

19347 

768345 

77563 

76463 

81731 

563875 

83563 

34698 

29342 

547427 

54973 

17179 

75659 

945956 

38137 

93965 

35446 

165675 

54246 

81367 

98237  - 

756431 

95864 

29787 

12845 

354747 

48135 

79826 

87677 

543864 

37975 

31275 

23444 

567456 

48467 

59689 

39878 

621367 

(64.) 

(65.) 

(66.) 

(67.) 

768856 

576654 

•   987654 

9873785 

674387 

678456 

123456 

1239564 

978874 

754543 

876864 

7591074 

567678 

786567 

234246 

3517569 

568594 

964432 

765183 

8598674 

639678 

699678 

345927 

2513756 

669657 

978321 

654678 

3454210 

594886 

6^8789  • 

456432 

7656754 

695756 

564673 

345719 

5467856 

789568 

895437 

765391 

5645781 

689689 

569128 

673123 

7893344 

638786 

678982 

437987 

3216677 

675968 

869771 

566789 

4569911 

958789 

668339 

544321 

6543344 

769896 

956234 

891389 

9576677 

153674 

195876 

219721 

1539900 

331767 

957412 

625247 

6662233 

355989  ■ 

573375 

431321 

4235566 

11522492 

13046667 

9945448 

99796675 

SUBTEACTIOK". 


29 


SUBTEAOTIOK 

42 .  Subtraction  is  the  process  of  finding  the  differ- 
ence between  two  numbers  of  the  same  unit  value. 

43.  The  Difterence  or  Remainder  is  the  result  ob-= 
tained. 

SuBTEACTioi^  Table. 


1  from   2  leaves 

1 

2  from   3  leaves    1 

3  from   4  leaves 

1 

4  from  5  leaves   1 

1  from    3  leaves 

2 

2  from   4  leaves    2 

3  from    5  leaves 

2 

4  from   6  leaves   3 

1  from   4  leaves 

3 

2  from    5  leaves    3 

3  from    6  leaves 

3 

4  from   7  leaves   3 

1  from   5  leaves 

4 

2  from   6  leaves   4 

3  from    7  leaves 

4 

4  from   8  leaves   4 

1  from   6  leaves 

5 

2  from    7  leaves    5 

3  from   8  leaves 

5 

4  from   9  leaves   5 

1  from    7  leaves 

6 

2  from    8  leaves    6 

8  from    9  leaves 

6 

4  from  10  leaves   6 

1  from   8  leaves 

7 

2  from   9  leaves   7 

3  from  10  leaves 

7 

4  from  11  leaves    7 

1  from   9  leaves 

8 

2  from  10  leaves   8 

3  from  11  leaves 

8 

4  from  12  leaves   8 

1  from  10  leaves 

S 

2  from  11  leaves    9 

3  from  12  leaves 

9 

4  from  13  leaves    9 

1  from  11  leaves 

10 

2  from  12  leaves  10 

3  from  13  leaves 

10 

4  from  14  leaves  10 

5  from    6  leaves 

1 

6  from   7  leaves    1 

7  from   8  leaves 

1 

8  from   9  leaves    1 

5  from   7  leaves 

2 

6  from    8  leaves    2 

7  from    9  leaves 

2 

8  from  10  leaves    2 

5  from   8  leaves 

3 

6  from   9  leaves   3 

7  from  10  leaves 

3 

8  from  11  leaves'   3 

5  from   9  leaves 

4 

G  from  10  leaves   4 

7  from  11  leaves 

4 

8  from  12  leaves   4 

5  from  10  leaves 

5 

6  from  11  leaves   5 

7  from  12  leaves 

5 

8  from  13  leaves   5 

5  from  11  leaves 

6 

6  from  12  leaves   6 

7  from  13  leaves 

6 

8  from  14  leaves   6 

5' from  12  leaves 

7 

6  from  13  leaves   7 

7  from  14  leaves 

7 

8  from  15  leaves    7 

5  from  13  leaves 

8 

6  from  14  leaves   8 

7  from  15  leaves 

8 

8  from  16  leaves   8 

5  from  14  leaves 

9 

6  from  15  leaves   9 

7  from  16  leaves 

9 

8  from  17  le  ives   9 

5  from  15  leaves  10 

6  from  16  leaves  10 

7  from  17  leaves 

10 

8  from  18  leaves  10 

9  from  10  leaves 

1 

10  from  11  leaves    1 

11  from  12  leaves 

1 

12  from  13  leaves    1 

9  from  11  leaves 

2 

10  from  12  leaves    2 

11  from  13  leaves 

2 

12  from  14  leaves    2 

9  from  12  leaves 

3 

10  from  13  leaves    3 

11  from  14  leaves 

3 

12  from  15  leaves    3 

9  from  13  leaves 

4 

10  from  14  leaves    4 

11  from  15  leaves 

4 

12  from  16  leaves    4 

9  from  14  leaves 

5 

10  from  15  leaves    5 

11  from  16  leaves 

5 

12  from  17  leaves    5 

9  from  15  leaves 

6 

10  from  16  leaves    6 

11  from  17  leaves 

6 

12  from  18  leaves    6 

9  from  16  leaves 

7 

10  from  17  leaves    7 

11  from  18  leaves 

7 

12  from  19  leaves    7 

9  from  IT  leaves 

8 

10  from  IS  leaves    8  11  from  19  leaves 

8 

12  from  20  leaves    8 

9  from  18  leaves 

9 

10  from  19  leaves    9  11  from  20  leaves 

9  12  from  21  leaves    9 

9  from  19  leaves 

10 

10  from  20  leaves  10  11  from  21  leaves 

10  12  from  22  leaves  10 

30  simplekumbers. 

Mental  Exercises. 

1.  A  grocer  liaving  20  boxes  of  lemons,  sold  12  boxes; 
liow  many  boxes  had  lie  left  ? 

Analysis. — He  had  left  the  difference  between  20  boxes  and  12 
boxes,  which  is  8  boxes. 

2.  If  a  man  earn  12  dollars  a  week,  and  spend  7  for 
provisions,  liow  many  dollars  has  he  left  ? 

3.  If  I  borrow  15  dollars,  and  pay  9  dollars,  how  many 
dollars  remain  unpaid  ? 

4.  John  had  11  marbles,  and  lost  5  of  them;  how  many 
had  he  left  ? 

5.  From  a  cistern  containing  22  barrels  of  water,  9 
barrels  leaked  out;  how  many  barrels  remained  ? 

6.  In  a  school  are  24  boys  and  12  girls;  how  many 
more  boys  than  girls  ? 

7.  From  a  piece  of  cloth  containing  17  yards,  8  yards 
were  cut;  how  many  yards  remained? 

8.  Orin  paid  15  dollars  for  a  coat,  and  9  dollars  for  a 
pair  of  pantaloons ;  how  much  more  did  he  pay  for  the 
coat  than  for  the  pantaloons  ? 

9.  Cora  is  23  years  old,  and  her  brother  is  10  years 
younger;  how  old  is  her  brother  ? 

10.  A  jeweler  bought  a  watch  for  11  dollars,  and  sold 
it  for  18  dollars;  how  much  did  he  gain  ? 

11.  A  boy  gave  21  cents  for  some  pictures,  which  were 
worth  no  more  than  17  cents;  how  much  more  than  their 
value  did  he  give  for  them  ? 

12.  A  grocer  bought  a  barrel  of  sugar  for  16  dollars, 
but  not  proving  as  good  as  he  expected,  he  sold  it  for  11 
dollars;  what  did  he  lose  on  it  ? 


SUBTRACTION". 


31 


Promiscuous  Subtractio:n"  Table. 


5  from 
5  from 
9  from 
G  from 

7  from 
9  from 

5  from 

6  from 

8  from 

7  from 
2  from 
5  from 

9  from 
9  from 
7  from 
2  from 


14  how  many  ? 

9  how  many  ? 
10  how  many  ? 

7  how  many  ? 
12  how  many? 
12  how  many  ? 

10  how  many  ? 

11  how  many? 


9  how 
16  how 
11   liOW 

8  how 
14  how 
13  how 

9  how 
10  how 


many  r 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 


7 
8 
4 
7 
3 
5 
9 
8 


from 
from 
from 
from 
from 
from 
from 
from 


15  how 
17  how 

10  how 
14  how 

11  how 

13  how 
17  how 

14  how 


many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 


6  from 

8  from 

5  from 
from 
from 

9  from 

6  from 
8  from 


7 
3 


14  how 

15  how 
11  how 

10  how 
13  how 

11  how 

12  how 
10  how 


4  from 

3  from 

5  from 

7  from 

8  from 

9  from 

6  from 

4  from 


11  how 
10  how 

12  how 

13  how 

12  how 
16  how 

13  how 
12  how 


many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
manv  ? 
many  ? 
many  ? 

many? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many? 


8  from  16 

9  from  15 
7  from  11 

3  from  12 
6  from  15 
9  from  18 
6  from.  10 

4  from  13 


how  many  ? 
how  many  ? 
how  many? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 


how 


many 


44.  The  Minuend  is  the  number  to  be  subtracted  from, 

45.  The  Subtrahend  is  the  number  to  be  subtracted. 

46.  The  Sign  of  Subtraction  is  a  short  horizontal 
hue,  — ,  called  minus.  When  placed  between  two  num- 
bers, it  denotes  that  the  one  after  it  is  to  be  taken  from 
the  one  before  it.  Thus,  8  —  6=2,  is  read  8  minus  6 
equals  2,  and  shows  that  6,  the  suUraliend,  taken  from 
8,  the  minuend,  equals  2,  the  remainder. 


32  SIMPLER"  UMBERS. 


Case  I. 

47.  When  no  figure  in  the  subtrahend  is  greater 
than  the  corresponding  figure  in  the  minuend. 

1.  Erom  574  ta,ke  323. 

OPERATlOlf.  Analysis.      Write  tlie  less  number 

under  the  greater,  with  units  under 
Minuend,  Or±      m^jts,  tens  under  tens,  etc.,  and  draw  a 

Subtrahend,  23o      ^-^^^   underneath.      Then,  beginning  at 

Remainder,  251      the     right     hand,    subtract     separately 

each  figure  of  the  subtrahend  from  the 
figure  above  it  in  the  minuend.  Thus,  3  from  4  leaves  1,  which  is 
the  difference  of  the  units  ;  2  from  7  leaves  5,  the  difference  of  the 
tens  ;  3  from  5  leaves  2,  the  difference  of  the  hundreds.  Hence, 
we  have  for  the  whole  difference,  2  hundreds  5  teus  and  1  unit, 
V.X-  251. 

Examples  for  Pract.tce. 


Minuend, 

(1.) 

876 

(2.) 
349 

(3.) 
637 

(4.) 
608 

Subtrahend, 

,485 

212 

431 

104 

Remainder, 

441 

137 

206 

404 

(5.) 
987 

(6.) 
753 

(7.) 
438 

695 

647 

502 

421 

535 

340 

251 

17 

160 

From 

(9.) 

7642 

(10.) 
8730 

(11.) 
2369 

(12.) 

9786 

Take 

3211 

6430 

2104 

3126 

4431  2300  265  6660 


SUBTRACTION.  33 

Remainders. 

13.  From  4376  take  1254.  3122. 

14.  From  70342  take  50130.  20212. 

15.  From  137647  take  16215.  121432. 

16.  Subtract  32014  from  86325.  54311. 

17.  Subtract  2l7356  from  719568.  502212. 

18.  437615— 213502  ==liow  many?  224113. 

19.  732740  — 11520=hoYf  many?  721220. 

20.  2042674— 32142=:liow  many? 

21.  8461203  — 7161003  =how  many? 

22.  From  three  thousand  two  hundred  seventy-six,  take 
two  thousand  one  hundred  forty-three. 

23.  From  one  hundred  eighty-three  thousand  four  hun- 
dred sixty,  take  fifty-two  thousand  one  hundred  fifty. 

Ans.  131310. 

24.  A  man  bought  a  piece  of  property  for  7634  dollars, 
and  sold  the  Ga-^ne  for  3132  dollars;  what  did  he  lose  ? 

Afis.  4502  dollars. 

25.  A  merchant  sold  goods  to  the  amount  of  41763  dol- 
lars, and  by  so  doing  gained  11521  dollars  ;  what  did  the 
goods  cost  him  ?  A7is.  30242  dollars. 

26.  A  drover  bought  3245  sheep,  and  sold  1249  of 
them;  how  many  sheep  had  he  left? 

27.  A  general  before  commencing  a  battle  had  18765 
men  in  his  army  ;  after  the  battle  he  had  only  8530  ;  how 
many  men  did  he  lose?  Ans.  10235. 

28.  Two  persons  bought  a  block  of  buildings  for  69524 
dollars;  one  paid  47321  dollars;  how  much  did  the  other 
pay  ?  Ans.  22203  dollars. 

29.  If  a  man's  annual  income  is  13460  dollars,  and  his 
expenses  are  3340  dollars,  what  does  he  save  ? 

Ans.  20120  dollars. 


34  SIMPLE    liTUMBEES. 

Case  II. 

48.  When  any  figure  in  the  subtrahend  is  greater 
than  the  corresponding  figure  in  the  minuend. 

I.  From  846  take  359. 

OPERATION.       Analysis.     Since  we  cannot  take  9  units  from 
m    ^.  6   units,  we  add  10  units  to   6   units,  making  16 

9  q-a  units  ;  and  9  units  from  16   units  leave  7  units, 

o  ^  g  But  as  we  have  added  10  units,  or  1  ten  to  the 

q  g  g  minuend,  we   shall   have  a  remainder  1    ten  too 

•  large,  to  avoid  which,  we  add  1  ten  to  the  5  tens 

^^  *  in  the   subtrahend,  making  6  tens.    We  can  not 

take  6  tens  from  4  tens  ;  so  we  add  10.  tens  to  4,  making  14  tens  ; 
6  tens  from  14  tens  leaves  8  tens.  Now,  having  added  10  tens,  or 
1  hundred,  to  the  minuend,  we  shall  have  a  remainder  1  hundred 
too  large,  unless  we  add  1  hundred  to  the  3  hundreds  in  the  subtra- 
hend, making  4  hundreds ;  4  hundreds  from  8  hundreds  leave  4 
hundreds,  and  we  have  for  the  total  remainder,  487. 

The  process  of  adding  10  to  the  minuend  is  sometimes  called  borrowing  10 ; 
and  that  of  adding  1  to  tne  next  figure  of  the  subtrahend,  carrying  one. 

49 .  From  the  preceding  example  and  illustration  we 
have  the  following  general 

KuLE.  I.  Write  the  less  number  under  the  greater, plac- 
ing units  of  the  same  order  in  the  same  column. 

II.  Beginning  at  the  right  hand^  take  each  figure  of  the 
subtrahend  from  the  figure  above  it,  and  lorite  the  result 
underneath, 

III.  If  any  figure  in  the  subtrahend  be  greater  than  the 
corresponding  figure  above  it,  add  10  to  that  upper  figure 
before  subtracting,  and  then  add  1  to  the  next  left-hand 
figure  of  the  subtrahend. 

Proof.  1st.  Add  the  remainder  to  the  subtrahend; 
the  sum  will  he  equal  to  the  minuend.     Or, 

2d.  Subtract  the  remainder  from  the  minuend ;  the 
difference  will  be  equal  to  the  subtrahend. 


SUBTRACTION. 


35 


Examples  for  Practice. 


Minuend,            753 

(2.) 
6731 

(3.) 
3248 

(4.) 
90361 

Subtrahend,       469 

2452 

1863 

6284 

Remainder,       284 

4279 

1385 

84077 

(5.) 

Miles. 

3146 

(6.) 

Bushels. 

19472 

(7.) 

Dollars. 

45268 

(8.) 

Feet. 

24760 

3529 

14681 

24873 

3478 

617 

4791 

20395 

21282 

(9.) 

Rods. 

40307 

(10.) 

Days. 

14605 

(11.) 

Acres. 

23617 

(12.) 

Gallons. 

980076 

38421 

8341 

14309 

94087 

1886 

6264 

9308 

885989 

(13.) 

Men, 

17380 

(14.) 

Sheep. 

282731 

(15.) 

Barrels. 

80014 

(16.) 

Tons. 

941000 

3417 

90756 

43190 

5007 

13963 

191975 

36824 

935993 

(17.) 

3077097 

(18.) 
3000001 

(19.) 
1970000 

1829164 

2199077 

1361111 

1247933 


800924 


608889 


■in, 


36 


SIMPLE     KUMBERS 


(20.) 

6000000 
999999 

5000001 


/ 


(21.) 
8000800 

457776 

7543024 


(22.) 
103810040 
91300397 
12509643 


Ans.  1975645. 
Ans,  4918889. 
Ans.  6939154. 
Ans.  891563. 
A71S.  999999. 


23.  234100  — 9970=:how  many  ?  A7is.  224130. 

24.  3749001— 349623  =:how  many? 

25.  4000320— 20142=how  many? 

26.  14601896  — 764059=:how  many? 

27.  From  4716359  take  2740714. 

28.  From  7867564  take  2948675. 

29.  From  7788996  take  849842. 

30.  From  1073563  take  182000. 

31.  From  1111111  take  111112. 

32.  Subtract  1234509  from  8643587.    A71S.  7409078. 

33.  Subtract  1000  from  1100000.  Ans.  1099000. 

34.  Subtract  100701  from  846587. 

35.  Subtract  432986702100  from  539864298670.  V, 

36.  Subtract  29176807982  from  86543298765.  r^  c-,  : , 

37.  A  speculator  bought  wild  lands  for  10580  dollars, 
and  sold  tliem  for  7642  dollars  ;  how  much  did  he  lose  ? 

Ans.  2938  dollars.| 

38.  Napoleon  the  Great  was  born  in  1769,  and  died  in 
1821  ;  how  old  was  he  at  his  death  ?        Ans.  52  years. 

39.  Gunpowder  was  invented  in  1330,  and  printing  in 
1440  ;  how  many  years  between  the  two  ?       A71S,  110. 

40.  George  Washington  was  born  in  1732,  and  died  in 
1799  ;  how  old  was  he  at  his  death?        Ans,  67  years. 

41.  The  first  newspaper  published  in  America  was 
issued  at  Boston  in  1704  ;  how  long  was  that  before  the 
death  of  Benjamin  Franklin,  which  occurred  in  1790  ? 

A71S.  86  years. 


PROMISCUOUS     EXAMPLES.  "  37 

42.  The  first  steamboat  in  the  United  States,  built  by 
Eobert  Fulton,  in  1807,  made  a  trip  from  New  York  to 
Albany  in  33  hours  ;  how  many  years  from  that  time  to 
the  visit  of  the  Great  Eastern  to  this  country  in  1860? 

Ans.  53  years. 

43.  Queen  Victoria  was  bom  in  1819  ;  what  was  her 
age  in  1862?  '  A71S,  43  years. 

44.  The  United  States  contain  2983153  square  miles, 
and  the  British  North  American  Provinces  3125401  square 
miles ;  how  many  square  miles  does  the  latter  country 
exceed  the  former  ?  A)is.  142248. 

Examples  Combinii^g  Addition  a]S"d  Subtraction. 

1.  A  farmer  having  450  sheep,  sold  124  at  one  time, 
and  96  at  another  ;  how  many  had  he  left  ?    Ans.  230. 

2.  If  a  man's  income  is  175  dollars  a  month,  and  he 
pays  25  dollars  for  rent,  44  dollars  for  provisions,  and  1 8 
dollars  for  other  expenses,  how  much  will  he  have  left  ^f 

A?is.  88  dollars.   . 

3.  A  man  gave  his  note  for  3245  dollars.  He  paid  at 
one  time  780  dollars,  and  at  another  484  dollars  ;  how 
much  remained  unpaid  ?  Ans.  1981  dollars. 

4.  A  man  paid  140  dollars  for  a  horse  and  165  dollars 
for  a  carriage.  He  afterward  sold  them  both  for  300  dollars; 
did  he  gain  or  lose, and  how  much  ?   Ans.  Lost  5  dollars. 

5.  A  flour  merchant  having  700  barrels  of  flour  on  hand, 
sold  278  barrels  to  one  man,  and  142  to  another;  how 
many  barrels  had  he  left  ?  A?is.  280  barrels. 

6.  Three  men  bought  a  farm  for  9840  dollars.  The 
first  paid  2672  dollars,  the  second  paid  3089  dollars,  and 
the  third,  the  remainder  ;  what  did  the  third  pay  ? 

Ans.  4079  dollars. 


38  SIMPLE     I^UMBERS. 

7.  A  man  bouglit  a  liouse  for  1500  dollars,  and  having 
expended  315  dollars  for  repairs,  sold  it  for  2000  dollars  ; 
wliat  was  liis  gain  ?  Ans.  185  dollars. 

8.  Henry  Jones  owns  property  to  the  amount  of  36748 
dollars, of  which  he  has  invested  in  real  estate  12850  dollars, 
in  personal  property  9086  dollars,  and  the  remainder  he  has 
in  bank;  how  much  has  he  in  bank  ?  Ans.  14812  dollars. 

9.  A  grocer  bought  275  pounds  of  butter  of  one  farmer, 
and  318^pounds  of  another  ;  he  afterward  sold  210  pounds 
to  one  customer,  and  97  to  another ;  how  man)  pounds 
had  he  left  ?  Ans.  286  pounds. 

10.  A  man  deposited  in  bank  10476  dollars ;  he  drew 
out  at  one  time  2356  dollai^s,  at  another  1242.  and  at  an- 
other 737  dollars  ;  hov/  much  had  he  remaining  in  bank? 

A71S,  6141  dollars. 

11.  Borrowed  of  my  neighbor  at  one  time  680  dollars, 
at  another  time  910  dollars,  and  at  another  time  218 
dollars.  Having  paid  him  1309 'dollars.  1  »w  much  do  I 
still  owe  him  ?  A?'iS.  499  dollars. 

12.  A  man  bought  3  lots  ;  for  the  first  he  paid  2480 
dollars,  for  the  second  3137  dollars,  and  for  the  third  as 
much  as  for  the  other  two  ;  he  afterward  sold  them  for 
15000  dollars  ;  what  was  his  gain  ?    A71S.  3766  dollars. 

13.  A  farmer  raised  1864  bushels  of  wheat*,  and  J129 
bushels  of  corn.  Having  sold  1340  bu&hels  of  wheat,  and 
1000  bushels  of  corn,  how  manv  busliels  of  each  has  he 
remaining  ?  A7is.  524  bushels,  and  129  bushels. 

14.  A  gentleman  worth  25800  dellors,  bequeatl'ed  his 
estate  so  that  each  of  his  two  sons  should  have  94  )0  dol- 
lars, and  his  daughter  the  remainder;  what  vas  tht 
daughter's  portion  ?  :  '  y  Cff^ 


MULTIPLICATIOJ^". 


39 


MULTIPLICATION". 

50.  Multiplication  is  the  process  of  taking  one  of  two 
given  numbers  as  many  times  as  there  are  units  in  the  other. 

51.  The  Product  is  the  result  obtained 

Multiplication  Table. 


Once 

1 

is 

1 

2  limes    1  are    2 

3  times    1  are    3 

4  times 

1  are    4 

Once 

2 

is 

2 

2  times    2  are    4 

3  times    2  are    6 

4  times 

2  are    8 

Once 

3 

is 

3 

2  tJmes    3  are    6 

3  times    3  are    9 

4  times 

3  are  12 

Once 

4 

is 

4 

2  times    4  are    8 

3  times    4  are  12 

4  times 

4  are  16 

Ones 

5 

is 

5 

2  times    5  are  10 

3  times    5  are  15 

4  limes 

5  are  20 

Once 

6 

ii 

C 

2  times    6  are  12 

3  times    G  are  18 

4  times 

6  are  24 

Once 

7 

is 

7 

2  times    7  are  14 

3  times    7  arc  21 

4  times 

7  aj  e  28 

Once 

8 

is 

8 

2  times    8  are  16 

3  tiroes    8  are  24 

4  times 

8  are  32 

Once 

9 

is 

9 

2  times    9  are  18 

3  times    9  are  2T 

4  times 

9  are  36 

Once 

10 

is 

10 

2  times  10  are  :^0 

3  times  10  are  30 

4  times 

10  are  40 

Once 

11 

is 

11 

2  times  11  are  22 

3  times  11  are  S3 

4  times 

11  are  44 

Once 

12 

is 

12 

2  times  12  are  24 

3  times  12  are  33 

4  t\rr.S£ 

12  arc  48 

5  limes 

1 

ra-e 

5 

g  times    1  are    6 

7  times    1  are    7 

8  times 

1  are    8 

5  times 

2 

are 

10 

6  times    2  are  12 

7  times    2  are  14 

8  times 

2  are  16 

5  times 

3 

are 

15 

6  times    3  are  IS 

7  times    3  are  21 

8  times 

3  are  24 

5  times 

4 

are 

20 

G  times    4  are  Si' 

7  times    4  are  28 

8  times 

4  are  32 

5  T.iia'^s 

5 

are 

25 

6  times    5  are  SO 

7  times    5  are  35 

8  times 

5  are  4:- 

5  limes 

6  are  30 

6  tiiues    6  ave  36 

7  times    6  are  42 

8  times 

6  are  4-. 

5  times 

7 

are 

05 

G  times    7  are  42 

7  times    7  are  49 

8  times 

7' arc  56    i 

5  tinies 

8 

are 

40 

G  times    8  are  4b 

7  time:i    3  are  56 

8  times 

8  are  64 

5  rimes 

9 

are 

45 

6  times    9  are  54 

7  timc:-.    V;  are  63 

8  times 

9  are  72 

5  times 

10 

are 

50 

6  times  10  are  CO 

7tlmt-  10  arc  70 

8  times 

?0  are  80 

5  lime;; 

11 

are 

55 

6  times  11  are  <x 

7  times  11  are  77 

8  times 

Ii  are  88 

5  timu-j 

13 

are 

60 

6  times  12  are  72 

7  times  12  are  84 

.4  .        .             -             1 

8  times  12  are  96    { 

9  times 

1 

are 

9; 

10  times   1  Pre   10 

11  times    1  are   11 

12  times 

1  are   12  | 

9  times 

2 

are 

18 

10  times    2  are    20  j 

11  times   2  are   22 

12  timps 

2  are   24 

9  times 

3 

are 

27 

10  times   3  are   30 

11  times   3  are   33  j 

12  times 

3  are   36  . 

9  times 

4 

are 

S6 

10  times   4  are   40 

11  times   4  are   44  j 

13  times 

4  are   48 

9  times 

5 

are 

4.  j 

10  times   5  are   50  j 

11  times   5  arc   55  | 

12  times 

5  arc   GO  ! 

9  times 

G  are 

54  1 

10  times   G  are   '0 

11  times   6  are   66  ! 

12  times 

G  are    72  j 

9  times 

"7 

arc 

63 ; 

10  times    '"are    70  ] 

11  times   7  are   77  | 

12  times 

7  are   84  i 

9  times 

S' 

ai'e 

72! 

10  times   8  are   l")  . 

11  times   8  are   88  { 

12  times 

8  are    96 

9  tircee 

9 

are 

81  i 

10  times   9  are   00  i 

li  times   9  arc   99  ! 

12  times 

G  are  108 

0   '■•".T.   .- 

.10 

arv^ 

80  ; 

10  times  10  are  10'3  ] 

11  times  10  are  110  ! 

12  times  10  are  UO  i 

^l 

are 

?9  \ 

lOtimeb  11  are  119 

11  limes  11  are  121  ; 

12  times 

n  are  132  ! 

.  -     .  . 

-'- 

are 

10^  1 

i 

10  timorf  12  are  120  . 

11  times  12  are  132  j 

12  times  12  are  144 

40  simple   kumbers. 

Mei^tal  Exeecises.    . 

1.  At  9  cents  a  pound,  what  will  7  pounds  of  sugar  cost  ? 

Analysis.  Since  one  pound  costs  9  cents,  7  pounds  will  cost 
7  times  9  cents,  or  63  cents.  Therefore,  at  9  cents  a  pound,  7 
pounds  of  sugar  will  cost  63  cents. 

2.  At  6  dollars  a  week,  what  will  8  weeks'  board  cost  ?  -'■  3 

3.  When  flour  is  7  dollars  a  barrel,  what  will  11  barrels  7  J) 
cost? 

4.  If  Eollin  can  earn  10  dollars  in  one  month,  how  much 
can  he  earn  in  4  months  ?  In  9  months  ?  In  11  months  ? 

5.  What  will  be  the  cost  of  12  pounds  of  coffee,  at  9 
cents  a  pound  ? 

6.  A  5  dollars  a  ton,  what  will  9  tons  of  coal  cost  ? 

7.  At  4  dollars  a  yard,  what  will  8  yards  of  cloth  cost  ? 

8.  If  a  pair  of  boots  cost  5  dollars,  what  will  be  the  cost 
of  3  pairs ?     Of  6  pairs?     Of  7  pairs ?     Of  11  pairs  ? 

9.  Since  12  inches  make  a  foot,howmany  inches  in  3  feet? 
In  5  feet?    In  7  feet  ?     In  9  feet  ?    In  12  feet? 

10.  At  five  cents  a  quart,  what  will  6  quarts  of  milk 
cost  ?     10  quarts  ?     11  quarts  ? 

11.  If  a  man  earn  8  dollars  in  a  week,  how  much  can  he 
earn  in  6  weeks?  In  7  weeks  ?  In  8  weeks  ?  In  9  weeks? 

12.  If  9  bushels  of  apples  buy  one  barrel  of  flour,  how 
many  bushels  will  be  required  to  buy  3  barrels?  5  barrels? 
7  barrels  ?    9  barrels  ? 

13.  If  4  men  can  do  a  piece  of  work  in  8  days,  how 
many  days  will  it  take  one  man  to  do  it  ?  ^ 

14.  If  7  men  can  build  a  wall  in  3  days,  how  long  will 
it  take  one  man  to  build  it? 

15.  If  a  barrel  of  potatoes  last  6  persons  3  weeks,  how 
many  weeks  will  it  last  one  person  ? 


MULTIPLICATION". 


41 


Promiscuous  Multiplicatiok  Table. 


2  times 

3  times 

4  times 
7  times 
9  times 
6  times 

4  times 

6  times 

7  times 

3  times 

8  times 

6  times 

5  times 

7  times 

6  times 

9  times 
3  times 

7  times 


liow  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 


8  are 

9  are 

8  are 

5  are 

4  are  how  many 
3  are  ' 

9  are 
9  are 

6  are 


2  times 
6  times 

4  times 
9  times 

5  times 

5  times 
9  times 

6  times 
8  times 


9  are  how 
5  are  how 
7  are  how 

3  are  how 

7  are  how 

8  are  how 
5  are  how 

4  are  how 
3  are  how 


7  are  how  many 
9  are  ' 

8  are 
6  are 
3  are  how  many '. 

6  are  how  many ' 

7  are  how  many  ? 

8  are' '^ 


4  are 


1.Jl\J  \\       X1.LCLLX   V 

how  many 
how  many  ? 
how  many  ? 
how  many  ? 
how  many  ? 
how  many" 
how  many 
how  many  ? 


many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many? 
many  ? 
many  ? 
many  ? 

many? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many  ? 
many? 

2  times  4  are  how  many  ? 

5  times  9  are  how  many  ? 
9  times  8  are  how  many  ? 

3  times  3  are  how  many  ? 

2  times  3  are  how  many  ? 
7  times  4  are  how  many  ? 
0  times  8  are  how  many  ? 

3  times  6  are  how  many  ? 

6  times  10  are  how  many  ? 


7  times 
4  times 
9  times 
4  times 
6  times 
2  times 

8  times 
4  times 

9  times 


7  are  how 

2  are  how 
9  are  how 

3  are  how 
9  are  how 
6  are  how 
5  are  how 

4  are  how 

8  are  how 


8  times  7  are  how  many  ? 
5  times  4  are  how  many  ? 
3  times  5  are  how  many  ? 
3  times  4  are  how  many  ? 
8  times  6  are  how  many  ? 

7  times  8  are  how  many  ? 
5  times  3  are  how  many  ? 
3  times  6  are  how  many  ? 

8  times  8  are  how  many  ? 

52.  The  Multiplicand  is  the  number  to  be  multiplied, 

53.  The  Multiplier  is  the  number  which  shows  how 
many  times  the  multiplicand  is  to  be  taken. 

54.  The  Factors  are  the  multiplicand  and  multiplier. 

55.  The  Sig-n  of  Multiplication  is  the  obhque  cross, 
X .    It  shows  that  the  numbers  connected  by  it  are  to  be 

multiplied  together;   thus,  9x6=54,  is  read  9  twics  6 
equals  54. 


42  SIMPLE     NUMBERS. 

1.  Factors  are  producers,  and  the  multiplicand  and  multiplier  are  called  factors 
because  they  produce  the  product. 

2.  Multiplication  is  a  short  method  of  performing  addition  when  the  numbers 
to  be  added  are  equal. 


Case  I. 

5G,  When  the  multipHer  consists  of  one  figure. 

1.  Multiply  374  by  6. 

OPEKATION.  Analysis.     In  this  example  it  is  re- 

^      .  '     quired  to  take  374  six  times.     If  we  take 

So-^        the  units  of  each  order  6  times,  we  shall 

CSS 

.  Q  ^  A       ^^^^  ^^^^  entire  number  6  times.     There- 

,,,.,.       '  ^       fore,    writing   the    multiplier    under   the 

Multiplier,  6  .. \  p  ,,  IX.    1-        J 

unit  ngure  or  the  multiplicand,  we  proceed 

Product,  2  2  44       as  follows:  6  times  4  units  are  24  units, 

which  is  2  tens  and  4  'units;  write  the 
4  units  m  the  product  in  units'  place,  and  reserve  the  2  tens  to  add 
to  tho  next  product ;  6  times  7  tens  are  42  tens,  and  the  two 
tens  reserved  in  the  last  product  added,  are  44  tens,  which  is  4 
hundreds  and  4  tens  ;  write  the  4  tens  in  the  product  in  tens'  place, 
and  reserve  the  4  hundreds  to  add  to  the  next  product ;  G  times  3 
hundreds  are  eighteen  hundreds,  and  4  hundreds  added  are  22  hun- 
dreds, which, being  written  in  the  product  in  the  places  of  hundreds 
and  thousands,  gives,  for  the  entire  product,  2244. 

57.  The  unit  value  of  a  number  is  not  changed  by  re- 
peating the  number.  As  the  multiplier  always  expresses 
times,  the  product  must  have  the  same  unit  value  as  the 
multiplicand.  But,  since  the  product  of  any  two  num- 
bers will  be  the  same,  whichever  factor  is  taken  as  a  mul- 
tiplier, either  factor  may  be  taken  for  the  multiplier  or 
multiplicand. 

In  multiplying,  learn  to  pronounce  the  partial  results,  as  in  addition,  without 
naming  the  numbers  separately.  Thus,  in  the  last  example,  instead  of  saying 
6  times  4  are  24,  6  times  7  are  42  and  2  to  carry  are  44,  6  times  3  are  18  and  4  to 
carry  are  22 ;  pronounce  only  the  results,  24,  44,  22,  performing  the  operations 
mentally.    This  will  greatly  facilitate  the  process  of  multiplying. 


MULTIPLICATION. 


43 


Examples  foe  Practice. 


(2.) 

(3.) 

(4.) 

(5.) 

Multiplicand, 

842 

625 

718 

937 

Multiplier, 

4 

6 

7 

3 

Product, 

3368 

3750 

5026 

2811 

(6.) 

(7.) 

(8.) 

(9.) 

4328 

5073 

1869 

3265 

8 

5 

4 

9 

34624 

25365 

*      7476 

29385 

(10.) 

(11.) 

(12.) 

(13.) 

7186 

9010 

4079 

6394 

3 

7 

6 

8 

< 

^558 

63070 

24474 

51152 

(14.) 

(15.) 

- 

(16.) 

340071 

760892 

1976230 

2 

4 

5 

680142 


3043568 


9881150 


17.  Multiply  473126  by  9.  Ans, 

18.  Multiply  30789167  by  7.  A7is, 

19.  Multiply  87231420  by  8.  Ans. 

20.  What  will  be  tbe  cost  of  9380  busliels  of  wheat,  at 
9  shillings  a  bushel?  Ans.  84420  shillings. 

21.  What  will  be  the  cost  of  4738  tons  of  coal,  at  4  dol- 
lars a  ton?  Ans.  18952  dollars. 

22.  In  one  mile  are  5280  feet  ;  how  many  feet  in  8 
miles?  Ans.  42240  feet. 


/: 


44 


SIMPLE     NUMBEES. 


Case  II. 

5S»  When  the  multiplier  consists  of  two  or  more 
figures. 

1.  Multiply  746  by  23. 


Multiplicand, 
Multiplier, 


Product, 


OPERATION. 
746 

23 


times  the  mul- 
tiplicand, 
times  the  mul- 
tiplier. 

1^158  23]  Jj-^them-l- 


2238   3' 
1492    20 


Analysis.  "Writing 
fhe  multiplicand  and  mul- 
tiplier ag  in  Case  I,  first 
multiply  each  figure  in 
tlie  multiplicand  by  the 
unit  figure  of  the  multi- 
tiplier,  precisely  as  in 
Case  I.  Then  multiply 
by  the  2  tens.  2  tens  times  6  units,  or  6  times  2  tens,  are  12  tens, 
equal  to  1  hundred  and  2  tens  ;  place  the  2  tens  under  the  tens  fig- 
ure in  the  product  already  obtained,  and  add  the  1  hundred  to  the 
next  hundreds  prod'.iced.  2  tens  times  4  tens  are  8  hundreds,  and 
the  1  hundred  of  the  last  product  added  are  9  hundreds  ;  write  the 
9  in  hundreds'  place  in  the  product.  2  tens  times  7  hundreds  are 
14  thousands,  equal  to  1  ten  thousand  and  4  thousands,  which  we 
write  in  their  appropriate  places  in  the  product.  Then  adding  the 
two  partial  i^roducts,  we  have  for  the  entire  product,  17158. 

Hence  the  following  general 

EuLE.  I.  JVrite  the  ynultiplier  under  the  multiplicand, 
placing  units  of  the  same  order  under  each  other. 

II,  Multiply  the  multiplicand  iy  each  figure  of  the  mul- 
tiplier successively,  heginning  luith  the  unit  figure,  and 
write  the  first  figure  of  each  partial  product  under  the  fig- 
ure of  the  multiplier  used,  luriting  down  and  carrying  as 
in  addition. 

III.  If  there  are  partial  products,  add  them,  and  theiy 
Bum  loill  le  the  product  required. 


MULTIPLICATION". 


45 


Pkoof.  Multiply  the  multiplier  by  the  multiplicand, 
and  if  the  product  is  the  same  as  the  first  result,  the  work 
is  correct. 

When  the  multiplier  contains  two  or  more  figures,  the  several  results  obtained 
by  multiplying  by  each  figure  are  called  partial  products. 

Examples  for  Peactice. 


(1-) 

(2.) 

(3.) 

34732 

66784 

.  34075 

14 

24 

36 

138928 
34732 

486248 


227136 
113568 

1362816 


4.  Multiply  177242  by  19. 

5.  Multiply  1429689  by  55. 

6.  Multiply  364111  by  56. 

7.  Multiply  78540  by  95. 

8.  Multiply  6555  by  39. 

9.  Multiply  76419  by  17. 

10.  Multiply  26517  by  45. 

11.  Multiply  108336  by  58. 

12.  Multiply  209402  by  72. 

13.  Multiply  342516  by  56. 


204450 
102225 

1226700 
Ans,    3367598. 

Ans,  20390216. 

Ans.  7461300. 

Ans.  255645. 

Ans.  1299123. 

Ans.  1193265. 

Ans.  6283488. 

Ans.  15076944. 

Ans.  19180896. 

Ans.  36678288. 


14.  Multiply    764131  by  48. 

15.  There  are  52  weeks  in  a  year  ;  how  many  weeks  in 
1861  years  ?  Ans.  96772  weeks. 

16.  An  army  of  5746  men  ha\dng  plundered  a  city,  took 
so  much  money  that  each  man  received  37  dollars;  how 
much  money  was  taken  ?  Ans.  212602  dollars. 

17.  If  it  cost  47346  dollars  to  build  one  mile  of  railroad^ 
what  will  it  cost  to  build  76  miles  ? 

Ans.  3598296  dollars. 


46 


SIMPLE  NUMBERS. 

(18.) 

(19.) 

(20.) 

47696 

560341 

243042 

144 

304 

265 

190784 

2241364 

1215210 

190784 

1681023 

1458252 

47696 

6868224 

170343664 

486084 
64406130 

Ans.  15210774. 
Ans.  69371964. 
Ans,  4712394711. 
Ans,  2472748875. 


Ans,   7614328386. 


21.  Multiply      45678  by    333. 

22.  Multiply    202842  by    342. 

23.  Multiply  9636799  by    489. 

24.  Multiply  3064125  by    807. 

25.  Multiply  5610327  by  2034. 

26.  Multiply  1900731  by  4006. 

27.  A  gentleman  bought  307  horses  for  shipping,  at  the 
rate  of  105  dollars  each ;  how  much  did  he  pay  for  the 
whole  ? 

28.  What  will  be  the  value  of  976  shares  cf  railroad 
stock,  at  98  dollars  a  share  ?  A71S.  95648  dollars. 

29.  A  man  bought  48  building  lots,  at  1236  dollars  each ; 
what  did  they  all  cost  him  ?  Ans.  59328  dollars. 

30.  How  many  yards  of  broadcloth  in  487  pieces,  each 
piece  containing  37  yards?  Aiis.  18019  yards. 

31.  If  it  require  135  tons  of  iron  for  one  mile  of  railroad, 
how  many  tons  will  be  required  for  196  miles  ? 

Ans.  26460  tons. 

32.  How  many  oranges  in  356  boxes,  each  box  contain- 
ing 264  oranges  ?  Ans.  93984  oranges. 

33.  If  it  require  6894  shingles  for  the  roof  of  a  house, 
how  many  sliingles  will  be  required  for  19  such  houses? 


MULTIPLICATIOlvr,  4? 

34.  37896  x  149=how  many?  Ans.        5646504. 

35.  8567  X  462=liow  many  ?  Ans.        3957954. 

36.  6793  X  842=how  many?  Ans.        5719706. 

37.  674200  x  2104=liow  many  ?      Ans,  141851 6800. 

38.  15607  X  3094==  how  many  ?        Ans.      48288058. 

39.  83209  X  4004=liow  many  ? 

40.  Multiply  31416  by  175. 

41.  Multiply  40930  by  779.  Atis.  31884470. 

42.  Multiply  4567  by  9009.  A?is.  41144103. 

43.  Multiply  7071  by  556.  Ans.     3931476. 

44.  Multiply  291042  by  125.  Ans.  36380250. 

45.  Multiply  54001  by  5009. 

46.  Multiply  twelve  thousand  thirteen,  by  twelve  hun- 
dred four.  Ans.  14463652. 

47.  Multiply  thirty-seven  thousand  seven  hundred 
ninety-six,  by  four  hundred  eight. 

48.  Multiply  one  million  two  hundred  forty-six  thousand 
eight  hundred  fifty-three,  by  nine  thousand  seven. 

Ans.   11230404971. 

49.  What  will  be  tho  cost  of  building  128  miles  of  rail- 
road, at  6375  dollars  per  mile  ?      Ans.  816000  dollars. 

50.  A  crop  of  cotton  was  put  up  in  126  bales,  each  bale 
containing  572  pounds  ;  what  was  the  weight  of  the  entire 
crop  ?  Ans.  72072  pounds. 

51.  Two  towns,  243  miles  apart,  are  to  be  connected  by 
a  railroad,  at  a  cost  of  39760  dollars  a  mile  ;  what  will 
be  the  entire  cost  of  the  road  ?    Ans.  9661680  dollars. 

52.  Allowing  an  acre  of  land  to  produce  105  bushels,  how 
much  would  246  acres  produce  ?     Ans.  25830  bushels. 

53.  If  a  garrison  of  soldiers  consume  5789  pounds  of 
bread  a  day,  how  much  will  they  consume  in  287  days  ? 

Ans.  1661443  pounds. 


48 


SIMPLE     KUMBEES. 


contkactioi^s. 

Case  L- 

59,  When  the  multipher  is  a  composite  number. 

A  Composite  Number  is  one  that  may  be  produced 
by  multiplying  together  two  or  more  numbers ;  thus,  18 
is  a  composite  number,  since  6  x  3=18  ;  or,  9  x  2=18  ; 
or,  3x3x2  =  18. 

00.  The  Component  Factors  of  a  number  are  the 
several  numbers  which,  multiplied  together,  produce  the 
given  number ;  thus,  the  component  factors  of  20  are  10 
and  2,  (10  x  2=20)  ;  or,  4  and  5,  (4  x  5=20)  ;  or,  2  and 
2  and  5,  (2x2x5  =  20). 

The  pupil  must  not  confound  the  factors  with  the  pm-ts  of  a  number. 
Thus,  ihe  factors  of  which  twelve  is  composed  are  4  and  3,  (4x3=12) ;  while  the 
parts  of  which  12  is  composed  are  8  and  4,  (8  +  4=12),  or  10  and  2,  (10  +  2=12). 
Th(j  factors  are  multiplied^  while  the  parts  are  added.,  to  produce  the  number, 

1.  What  will  32  horses  cost,  at  174  dollars  apiece  ? 

OPEKATiON.  Aif ALYSis.    The  fac- 


Multiplicand, 
Ist  factor, 

2d  factor, 
Product, 


174  cost  of  1  horse. 

4 

696  cost  of  4  horses. 
8 


tors  of  32  are  4  and  8. 
If  we  multiply  the  cost 
of  1  liorse  by  4,  we  ob- 
tain the  cost  of  4  horses ; 
and  by  multiplying  the 
cost  of  4  horses  by  8, 
we  obtain  the  cost  of 


5568  cost  of  32  horses. 
8  times  4  horses,  or  32  horses,  the  number  bought. 

61.  Hence  the  following 

EuLE.  I.  Separate  the  composite  numler  into  two  or 
more  factors. 

II.  Multiply  tlie  multiplicand  ly  one  of  these  factors,  and 
the  product  hy  another,  and  so  on  until  all  the  factors  have 
teen  used  ;  the  last  product  loill  he  the  product  required. 

The  product  of  any  number  of  factors  will  be  the  same  in  whatever  order  Ihey 
are  multiplied.    Thus.  4  x  3  x  5=60,  and  5  x  4  x  3=60. 


1 


MULTIPLICATION.  49 


Examples  foe  Practice. 


1.  Multiply  521  by  16=4  x  4.  Ans.  8336. 

2.  Multiply  4350  by  25  =  5  x  5.  Ans.  108750. 

3.  Multiply  10709  by  36  =  6  x  6.  Ans.  385524. 

4.  Multiply  21700  by  27=3  x  9.  A?is.  585900. 

5.  Multiply  783473  by  42  =  6  x  7.      Ans.  32905866. 

6.  Multiply  764131  by  48  =  6  x  8.      Ans.  36678288. 

7.  Multiply  40567  by  96=8  x  12.        Ans.  3894432. 

8.  Multiply  182642  by  120=4  x  5  x  6. 

Ans.  21917040. 
•     9.  Multiply  20704  by  84=3  x  4  x  7.    A?is.  1739136. 

10.  Multiply  564120  by  140=4  x  5  x  7. 

Ans.  78976800. 

11.  What  will  56  acres  of  land  cost,  at  147  dollars  an 
acre  ?  Ans.  8232  dollars. 

12.  What  will  75  yoke  of  cattle  cost,  at  184  dollars  a 
yoke  ?  Ans.  13800  dollars. 

13.  If  a  ship  sail  380  miles  a  day,  how  far  will  she  sail 
in  45  days  ?  Ans.  17100  miles. 

14.  What  is  the  value  of  3426  pounds  of  butter,  at 
18  cents  a  pound  ?  Ans.  61668  cents. 

y-  15.  What  will  be  the  cost  of  125  horses,  at  208  dollars 

each?  Ans.  26000  dollars. 

« 

16.  What  is  the  value  of  1342  acres  of  land,  at  28  dol- 
lars an  acre  ? 

17.  What  will  be  the  cost  of  28  pieces  of  broadcloth, 
each  piece  containing  42  yards,  at  4  dollars  a  yard  ? 

Ans.  4704  dollars. 

18.  What  will  be  the  cost  of  16  sacks  of  coffee,  each 
sack  containing  75  pounds,  at  9  cents  a  pound  ? 

Ans.  10800  cents. 
8 


50  simple    numbers. 

Case  II. 

62.  When  tlie  multiplier  is  10,  100,  1000,  etc. 

If  we  annex  a  cipher  to  the  multiplicand,  each  figure  is 
removed  one  place  toward  the  left,  and  consequently  the 
value  of  the  whole  number  is  increased  tenfold.  If  two 
ciphers  are  annexed,  each  figure  is  removed  two  places 
toward  the  left,  and  the  value  of  the  number  is  increased 
one  hundredfold ;  and  every  additional  cipher  increases 
the  value  tenfold. 

Hence  the  following 

Rule.  Annex  as  many  cipliers  to  the  multiplicand  as 
there  are  ciphers  in  the  multiplier  j  the  number  so  formed 
is  the  product  required. 

Examples  for  Practice. 

1.  Multiply  246  by  10.  Jns,  2460. 

2.  Multiply  97  by  100.  Ans.  9700. 

3.  Multiply  1476  by  1000.  A7is.    1476000. 

4.  Multiply  7361  by  10000.  A7is.  73610000. 

5.  At  47  dollars  an  acre,  what  wiU  10  acres  of  land  cost  ? 

Ans.  470  dollars. 

6.  What  will  be  the  cost  of  100  horses,  at  95  dollars  a 
head  ?  Ans.  9500  dollars. 

7.  What  will  be  the  cost  of  1000  fruit  trees,  at  18  cents 
apiece  ?  Ans.  18000  cents. 

8.  If  one  acre  of  land  produce  28  bushels  of  wheat, 
how  many  bushels  will  100  acres  produce  ?   Ans.  2800. 

9.  If  a  man  save  386  dollars  a  year,  what  will  he  save 
in  10  years  ?  A^is.  3860  dollars. 

10.  If  the  freight  on  a  barrel  of  flour  from  Chicago  to 
New  York  is  47  cents,  what  will  it  be  on  100000  barrels  ? 

A71S.  4700000  cents. 


MULTIPLICATION.  51 

Case  III. 

63.  When  there  are  ciphers  at  the  right  hand  of 
one  or  both  of  the  factors. 

1.  Multiply '7200  by  40. 

OPERATION.  Analysis.     The  multiplicand,  fac- 

Multiplicand,      7200  tored,  is  equal  to  72  x  100  ;  the  mul- 

Multiplier,            40  tiplier,  factored,  is  equal  to  4x10; 

— and  as  these   factors  taken    in   any 

Product,  /COoUUU  order  will  give  the  same  product,  we 

first  multiply  72  by  4,  then  this  product  by  100  by  annexing  two 
ciphers,  and  this  product  by  10  by  annexing  one  cipher.  Hence, 
the  following 

EuLE.  Multiply  the  significant  figures  of  the  multijM- 
cand  ly  those  of  the  multijMer,  and  to  the  product  annex 
as  many  ciphers  as  there  are  ciphers  on  the  right  of  either 
or  loth  factors. 

Examples  for  Practice. 

(1.)  (2.)  (3.) 

Multiply      3900-  1760  37200 

By  8000  3500        .  730000 

31200000  880  1116 

528  2604 


6160000  27156000000 

4.  Multiply  7080  by  164000.  Ans,    1152920000. 

5.  Multiply  27600  by  48000.  Ans.    1324800000. 

6.  Multiply  403700  by  30200.         Ans.  12191740000. 

7.  At  150  dollars  an  acre,  what  will  be  the  cost  of  500 
acres  of  land?  Ans.  75000  dollars. 

8.  What  will  be  the  freight  on  4000  barrels  of  flour,  at 
50  cents  a  barrel  ?  Ans.  200000  cents. 

9.  If  there  are  560  shingles  in  a  bunch,  how  many  shin- 
gles in  26000  bunehes  ?  Ans.  14560000. 


52  simple    numbeks. 

Examples  Combii^ing  Addition",   Subteactiok,  an"d 

Multiplication". 

1.  Bought  9  cords  of  wood  at  3  dollars  a  cord,  and  15 
tons  of  coal  at  5  dollars  a  ton  ;  what  was  the  cost  of  the 
wood  and  coal  ?  A7is.  102  dollars. 

2.  A  grocer  bought  6  tubs  of  butter,  each  containing 
64  pounds,  at  14  cents  a  pound :  and  4  cheeses,  each 
weighing  42  pounds,  at  8  cents  a  pound ;  what  was  the 
cost  of  the  butter  and  cheese  ?  A7is.  6720  cents. 

3.  If  a  clerk  receive  540  dollars  a  year  salary,  and  pay 
180  dollars  for  board,  116  dollars  for  clothing,  58  dollars 
for  books,  and  75  dollars  for  other  expenses,  how  much  will 
he  have  left  at  the  close  of  the  year?   Ans.  Ill  dollars. 

4.  A  farmer  having  2150  dollars,  bought  536  sheep  at 
2  dollars  a  head,  and  26  cows  at  23  dollars  a  head  ;  how 
much  money  had  he  left  ?  Ans.  480  dollars. 

5.  A  man  sold  three  horses  ;  for  the  first  he  received 
275  dollars,  for  the  second  87  dollars  less  than  for  the  first, 
and  for  the  third  as  much  as  for  the  other  two  ;  what  did 
he  receive  for  the  third  ?  Ans.  463  dollars. 

6.  Bought  76  hogs,  each  weighing  416  pounds,  at 
7  cents  a  pound,  and  sold  the  same  at  9  cents  a  pound  ; 
what  was  gained  ?  Ans.  63232  cents.     ^ 

7.  A  man  bought  14  cows  at  26  dollars  each,  4  horses 
at  112  dollars  each,  and  125  sheep  at  3  dollars  each ;  he 
sold  the  whole  for  1237  dollars  ;  did  he  gain  or  lose,  and 
how  much  ?  Ans.  Gained  50  dollars. 

8.  B  has  174  sheep,  C  has  three  times  as  many  lacking 
98,  and  D  has  as  many  as  B  and  C  together  ;  how  many 
sheep  has  D  ?  A^is.  598. 

9.  There  are  36  tubs  of  butter,  each  weighing  108 
pounds  ;  the  tubs  which  contahi  tlie  butter  each  weigh  19 


PROMISCUOUS     EXAMPLES.  53 

pounds ;  what  is  the  weight  of  the  butter  without  the 
tubs  ?  Ans.  3204  pounds. 

10.  A  man  paid  for  building  a  house  2376  dollars,  and 
for  his  farm  4  times  as  much  lacking  970  dollars  ;  what 
did  he  pay  for  both  ?  ^  : 

11.  A  merchant  bought  9  hogsheads  of  sugar  at  32  dol- 
lars a  hogshead,  and  sold  it  for  40  dollars  a  hogshead ; 
what  was  the  gain  ?  Ans,  72  dollars. 

12.  Bought  360  barrels  of  flour  for  2340  dollars,  and 
sold  the  same  at  8  dollars  a  barrel ;  what  was  gained  by 
the  bargain?  ^;i5.  540  dollars. 

13.  A  farmer  sold  462  bushels  of  wheat  at  2  d'ollars  a 
Imshel,  for  which  he  received  75  barrels  of  flour  at  9  dol- 
lars a  barrel,  and  the  balance  in  money ;  how  much 
money  did  he  receive  ?  Ans.  249  dollars. 

14.  Two  persons  start  from  the  same  point,  and  travel 
in  opposite  directions  ;  one  travels  at  the  rate  of  28  miles 
a  day,  the  other  at  the  rate  of  37  miles  a  day  ;  how  far 
apart  will  they  be  in  6  days  ?  A7is,  390  miles. 

15.  If  a  man  buys  40  acres  of  land  at  35  dollars  an 
acre,  and  56  acres  at  29  dollars  an  acre,  and  sell  the  whol^ 
for  32  dollars  an  acre,  what  does  he^ain  or  lose  ? 

— ^  Ans.  Gains  48  dollars. 

16.  In  an  orchard,  76  apple  trees  yield  18  bushels  of 
apples  each,  and  27  others  jdeld  21  bushels  each  ;  what 
are  the  apples  worth,  at  30  cents  a  bushel  ? 

Ans.  58050  cents.    . 

17.  A  man  bought  two  farms,  one  of  136  acres  at 
28  dollars  an  acre,  and  another  of  140  acres  at  33  dollars 
an  acre  ;  he  paid  at  one  time  4000  dollars,  and  at  another 
time  1875  dollars  ;  how  much  remained  unpaid  ? 

Ans.  2553  dollars. 


54 


SIMPLE     LUMBERS. 


DIYISIOK 

64.  Division  is  the  process  of  finding  lioiu  many  times 
one  number  is  contained  in  another  of  the  same  kind. 

65.  The  Quotient  is  the  result  obtained,  and  shows 
how  many  times  the  divisor  is  contained  in  the  dividend. 

Divisio:n"  Table. 


in 

2    2    times 

2 

in 

4 

2  times 

3 

in 

6    2  times 

in 

3    3    times 

2 

in 

6 

3  times 

3 

in 

9    3  times 

in 

4    4    times 

2 

in 

8 

4  times 

3 

in 

12    4  times 

in 

5    5    times 

2 

in 

10 

5  times 

3 

in 

15    5  times 

in 

6    6    times 

2 

in 

12 

6  times 

3 

in 

18    6  times 

in 

7    7    times 

2 

in 

14 

7  times 

3 

in 

21    7  times 

in 

8    8    times 

2 

in 

16 

8  times 

3 

in 

24    8  times 

1 

in 

9    9    times 

2 

in 

18 

9  times 

3 

in 

27    9  times 

4 

in 

8    2  times 

5 

in 

10 

2  times 

6 

in 

12    2  times 

4 

in 

13    3  times 

5 

in 

15 

3  times 

6 

in 

18    3  times 

4 

in 

16    4  times 

5 

in 

20 

4  times 

6 

in 

24    4  times 

4 

in 

20    5  times 

5 

in 

25 

5  times 

6 

in 

SO  .5  times 

4 

in 

24    6  times 

5 

in 

30 

6  times 

6 

in 

36    6  times 

4 

in 

28    7  times 

5 

in 

35 

7  times 

6 

in 

42    7  times 

4 

in 

32    8  times 

5 

in 

40 

8  times 

6 

in 

48    8  times 

4 

in 

36    9  times 

5 

in 

45 

9  times 

6 

in 

54    9  times 

7 

in 

14    2  times 

8 

in 

16 

2  times 

9 

in 

18    2  times 

7 

in 

21    3  times 

8 

in 

24 

3  times 

9 

in 

27    3  times 

7 

in 

23    4  times 

8 

in 

32 

4  times 

9 

in 

36    4  times 

7 

in 

35    5  times 

8 

in 

40 

5  times 

9 

in 

45    5  times 

7 

in 

42    6  times 

8 

in 

48 

6  times 

9 

in 

54    6  times 

7 

in 

49    7  times 

8 

in 

56 

7  times 

9 

in 

63    7  times 

7 

in 

56    8  times 

8 

in 

64 

8  time? 

9 

in 

72    8  times 

7 

in 

63    9  times 

8 

in 

72 

9  times 

9 

in 

81    9  times 

10 

in 

20    2  times 

in 

22 

2  times 

12 

in 

24    2  times 

10 

in 

30    3  times 

in 

33 

3  times 

12 

in 

36    3  times 

10 

in 

40    4  times 

in 

44 

4  times 

12 

in 

48    4  times 

10 

in 

50    5  times 

in 

55 

5  times 

12 

in 

60    5  times 

10 

in 

GO    6  times 

in 

6f5 

G  times 

12 

in 

72    6  times 

10 

in 

70    7  times 

in 

77 

7  times 

12 

in 

84    7  times  . 

10 

in 

80    8  times 

in 

88 

8  times 

12 

in 

96    8  times 

10 

in 

90    9  times 

in 

99 

9  times 

12 

in 

108    9  times 

1 

\- 


division.  66 

Mei^^tal  Exercises. 

1.  How  many  barrels  of  flour,  at  6  dollars  a  barrel,  can 

be  bousflit  for  30  dollars  ? 

Analysts.  Since  6  dollars  will  buy  1  barrel  of  flour,  30  dollars 
will  buy  as  many  barrels  as  6  dollars  is  contained  times  in  30  dol- 
lars, which  is  5  times.  Therefore,  at  6  dollars  a  barrel,  5  barrels  of 
flour  can  be  bought  for  30  dollars. 

2.  How  many  oranges,  at  4  cents  apiece,  can  be  bought 
for  28  cents  ? 

3.  How  many  tons  of  coal,  at  5  dollars  a  ton,  can  be 
bought  for  35  dollars  ? 

4.  When  lard  is  7  cents  a  pound,  how  many  pounds  can 
be  bought  for  49  cents  ?     For  63  cents  ?     For  84  cents  ? 

5.  If  a  man  travel  48  miles  in  6  hours,  how  far  does  he 
travel  in  one  hour  ? 

6.  At  3  cents  apiece,  how  many  lemons  can  be  bought 
for  24  cents  ?    For  30  cents  ?    For  36  cents  ? 

7.  If  you  give  55  cents  to  5  beggars,  how  many  cents 
do  you  give  to  each  ? 

8.  If  a  man  builds  42  rods  of  wall  in  7  days,  how  many 
rods  can  he  build  in  1  day  ? 

9.  At  4  dollars  a  cord,  how  many  cords  of  wood  can  be 
bought  for  20  dollars  ?     For  28  dollars ?    For  32  dollars? 

10.  A  farmer  paid  33  dollars  for  some  sheep,  at  3  dollars 
a  head  ;  how  many  did  he  buy  ? 

11.  At  7  cents  a  pound,  how  many  pounds  of  sugar 
can  be  bought  for  63  cents  ?     For  84  cents  ? 

12.  If  a  man  spends  5  cents  a  day  for  cigars,  how  many 
days  will  50  cents  last  him  ?     60  cents  ? 

13.  At  12  cents  a  pound,  how  many  pounds  of  coffee 
can  be  bought  for  48  cents  ?  For  72  cents  ?  For  96  cents  ? 
For  120  cents  ? 


56 


SIMPLE     NUMBERS. 


Promiscuous  Division"  Table. 


6 


6  m  36,  liow  many  times  ? 

7  in  42,  how  many  times  ? 
9  in  81,  how  many  times? 
5  in  35,  how  many  times  ? 

in  72,  how  many  times  ? 
in  27,  how  many  times  ? 
in  20,  how  many  times  ? 
in  54,  how  many  times  ? 

\n  32,  how  many  times  ? 

in  AF\    Tinw  mnnv  timpis? 


8 
9 
4 
6 


8 
5 
6 

8 
7 
3 

7 


4 

7 

5 

7 

4 

6  in 

9  in 

Gin 


;n35 


n 
in 
in 


m  dxj,  now  many  times  r 
in  45,  how  many  times  ? 
in  42,  how  many  times  ? 

how  many  times  ? 

how  many  times  ? 

how  many  times  ? 
.  ___  .^  how  many  times? 


now  many  times  r 
how  many  times  ? 


9  in  63,  how  many  times? 

6  in  12,  how  many  times  ? 

7  in  28,  how  many  times  ? 
4  in  16,  how  many  times? 

7  in  49,  how  many  times? 
4  in  36,  how  many  times? 

8  in  64,  how  many  times  r 
8  in  40,  how  many  times  r 

how  many  times  ? 
how  many  times  ? 


4  in  28,  how  many  times? 

8  in  32,  how  many  times  ? 

6  in  48,  how  many  times? 

9  in  45,  how  many  times? 
8  in  48,  how  many  times  ? 

7  in  56,  how  many  times  ? 
3  in  21,  how  many  times  ? 
6  in  54,  how  many  times  ? 


many  times  ? 

^  how  many  times  ? 

4  in  32,  how  many  times  ? 
6  in  24,  how  many  times  ? 
9  in  72,  how  many  times  ? 

5  in  10,  how  many  times  ? 

4  in    8,  how  many  times  ? 

5  in  20,  how  many  times  ? 
2  in  10,  how  many  times  ? 


how  many  times  r 
how  many  times  ? 
iU,  how  many  times  ? 
14,  how  many  times  ? 
24,  how  many  times  ? 
30,  how  many  times  ? 
36,  how  many  times  ? 
.  .„  30,  how  many  times  ? 

66.  The  Dividend  is  the  numucx  v.j  u^  vxxvxvxv^v^. 

6T.  the  Divisor  is  the  number  by  which  to  divide. 

68.  The  Sign  of  Division  is  a  short  horizontal  hne, 
with  a  point  above  and  one  below,  -j-^.  It  shows  that  the 
number  before  it  is  to  be  divided  by  the  number  after  it. 
Thus,  20-^4=5,  is  read,  20  divided  by  4\s  equal  to  5. 

Division  is  also  expressed  by  writing  the  dividend  above, 

and  the  divisor  below,  a  short  horizontal  line ; 

12 
Thus,  — -  =  4,  shows  that  12  divided  by  3  equals  4. 
o 


DIVISION".  57 

Case  I. 
69.  When  the  divisor  consists  of  but  one  figure. 

1.  How  many  times  is  4  contained  in  848  ? 

OPERATION.  Analysis.    After  writing  the  divisor  on 

Dividend,  the  left  of  the  dividend,  with  a  line  between 

Divisor,      4  )  848  them,  begin  at  the  left  hand  and  say  :  4  is 

Quotient  212  contained   in  8,   2  times,  and  as  8  in  the 

dividend  is  hundreds,  the  2  in  the  quotient 
must  be  hundreds  ;  therefore  write  2  in  hundreds'  place  under  the 
figure  divided.  4  is  contained  in  4,  1  time,  and  since  4  denotes 
tens,  write  1  under  it  in  tens'  place.  4  in  8,  2  times,  and  since  8 
is  units,  write  2  in  units'  place  under  it,  and  we  have  212  for  the 
entire  quotient. 

Examples  for  Practice. 

(2.)  (3.)  (4.) 

Divisor,     3  )936  Dividend,  2 )  4862  4)  48844 

312  Quotient,  2431  12211 

5.  Divide  9963  by  3.  Ans.    3321. 

6.  Di^dde  5555  by  5.  Ans.     1111. 

7.  Divide  68242  by  2.  Ans.  34121. 

8.  Divide  66666  by  6. 

AYhen  the  divisor  is  not  contained  in  the  first  figure  of 
the  dividend,  find  how  many  times  it  is  contained  in  the 
first  tivo  figures. 

9.  How  many  times  is  4  contained  in  2884  ? 

OPERATION.  Analysis.     As  we  cannot  divide  2  by  4,  say 

4  )  2884  4  is  contained  in  28,  7  times,  and  write  the  7  in 

771"  hundreds'   place  ;  then  4  is  contained  in  8,  3 

times,  which  write  in  tens'  place  under  the  fig- 
ure divided  :  and  4  is  contained  in  4,  1  time,  which  write  in  units' 
place  in  the  quotient,  and  we  have  the  entire  quotient,  721. 


58 


SIMPLE     KUMBEES. 


(10.) 
3 ) 2469 

823 


Examples  foe  Peactice. 

(11.) 
5 ) 3055 

611 


(12.) 

2 ) 148624 

74312 


Ans.  711. 
Ans.  9122. 
Ans.  8111. 


13.  Divide  4266  by  6. 

14.  Divide  36488  by  4. 

15.  Divide  72999  by  9. 

16.  Divide  21777  by  7. 

After  obtaining  the  first  figure  of  tbe  quotient,  if  the 
divisor  is  not  contained  in  any  figure  of  the  dividend, 
place  a  cipher  in  the  quotient,  and  prefix  this  figure  to 
the  next  one  of  the  dividend. 

To  2>rejix  means  to  place  before,  or  at  the  left  hand. 

17.  How  many  times  is  6  contained  in  1824  ? 

OPEEATiox.  Analysis.    Beginning  as  in  the  last  exam- 

6  )  1824  pies,  say,  6  is  contained  in  18,  3  times,  which 

TTT  write  in  hundreds'  place  in  the  quotient ;  then 

6  is  contained  in  2  no  times,  and  write  a  cipher 

(0)  in  tens'  place  in  the  quotient,  and  prefixing  the  2  to  the  4.  say 

6  is  contained  in  24,  4  times,  whicli  write  in  units'  place,  and  we 

have  304  for  the  entire  quotient. 

Examples  for  Practice. 


(18.) 

(19.) 

(20.) 

4)3228                        7)28357 

3  )  912246 

807 

4051 

304082 

21.  Divide  40525  by  5. 

Ans.     8105. 

22.  Divide  36426  by  6. 

A71S.     6071. 

23.  Divide  184210  by  2. 

Ans.  92105. 

24.  Divide  85688  by  8. 

Ans.  10711. 

25.  Divide  273615  by  3. 

Ans.  91205. 

DIVISION.  59 

After  diyiding  any  figure  of  tlie  diyidend,  if  there  be  a 
remainder,  prefix  it  mentally,  to  the  next  figure  of  the 
dindend,  and  then  divide  this  number  as  before. 

26.  How  many  times  is  4  contained  in  943  ? 

OPEPvATiON.  Analysis.    Here  4  is  contained  in  9,  2 

4 )  943  times,  and  there  is  1  remainder,  which 

^^  -p  prefix  mentally  to  the  next  figure,  4,  and 

ZoD  ..,6  Kem.  ^^^  ^  .g  contained  in  14,  3  times,  and  a  re- 
mainder of  2,  which  prefix  to  3,  and  say,  4  is  contained  in  23,  5 
times,  and  a  remainder  of  3.  This  3  which  is  left  after  performing 
the  last  division  should  be  divided  by  the  divisor  4 ;  but  the  method 
of  doing  it  cannot  be  explained  here,  and  so  we  merely  indicate  the 
division  by  placing  the  divisor  under  it ;  thus,  f .  The  entire  quo- 
tient is  written  235|,  which  may  be  read,  two  hundred  thirty -five 
and  thi^ee  divided  by  four,  or,  two  hundred  thirty-fl^  e  and  a  remain- 
der  of  three. 

Wheu  the  process  of  dividing  is  performed  mentally,  and  the  results  only  are 
•written,  as  in  the  preceding  examples,  the  operation  is  termed  S/W7't  Division. 

From  the  foregoing  examples  and  illustrations,  we  de- 
duce the  following 

^  EuLE.    I.  Write  the  divisor  at  the  left  of  the  dividend^ 
with  a  line  hetween  them. 

II.  Beginning  at  the  left  hand,  divide  each  figure  of  the 
dividend  dy  the  divisor,  and  write  the  result  tender  the 
dividend. 

III.  If  there  he  a  remainder  after  dividing  any  figure^ 
regard  it  as  prefixed  to  the  figure  of  the  next  lower  order 
in  the  dividend,  and  divide  as  before. 

lY.  Should  any  figure  or  part  of  the  divideiid  he  less 
than  the  divisor,  ivrite  a  cipher  in  the  quotient,  and  prefix 
the  numher  to  the  figure  of  the  next  lower  order  in  the  divi- 
dend, and  divide  as  hefore. 

V.  If  there  he  a  remainder  after  dividing  the  last  figure f 
place  it  over  the  divisor  at  the  right  hand  of  the  quotient. 


60 


SIMPLE    NUMBERS. 


Proof.  Multiply  the  divisor  and  quotient  together,  and 
to  the  product  add  the  remainder,  if  any  ;  if  the  result 
be  equal  to  the  dividend,  the  work  is  correct. 

1.  This  method  of  proof  depends  on  the  fact  that  division  is  the  reverse  of  mul- 
tiplication. The  dividend  answers  to  the  pi"Oduct,  the  divisor  to  one  of  the  fac- 
tors, and  the  quotient  to  the  other  factor. 

2.  In  multiplication  the  two  factors  are  given,  to  find  the  product :  in  division 
the  product  and  one  of  the  factors  are  given,  to  find  the  other  factor. 

Examples  for  Practice. 
1.  Divide  8430  by  6. 


OPERATIOlsr. 

PROOF. 

Divisor,      6  )  8430   Dividend. 

1405   Quotient. 

•4     A  /^  r^ 

6   Divisor. 

x4/}0    Quotient. 

- 

8430   Dividend. 

(2.)                                  (3.) 

(4.)    . 

5  )  730490                        7 )  510384 

8 ) 6003424 

146098                               72913 

750428 

Quotients.     . 

5.  Divide  87647  by  7. 

12521. 

6.  Divide  94328  by  8. 

11791. 

7.  Divide  43272  by  9. 

4808. 

8.  Divide  377424  by  6. 

62904. 

9.  Divide  975216  by  8. 

121902. 

10.  Divide  46375028  by  7. 

6625004. 

11.  Divide  4763025  by  9. 

529225. 

12.  Divide  42005607  by  7. 

6000801. 

13.  Divide  72000450  by  9. 

8000050. 

14.  Divide  97440643  by  8. 

12180080f. 

15.  Divide  65706313  by  9. 

7300701f 

16.  Divide  3627089  by  6. 

604514|. 

.  17.  Divide  4704091  by  7. 

672013. 

DIVISION.  61 

18.  Diyide  16344  dollars  equally  among  6  men  ;  what 
will  each  man  receive  ?  Ans.  2724  dollars. 

19.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  can 
be  bought  for  87605  dollars  ?  A7is.  12515  barrels. 

20.  In  one  week  there  are  7  days  ;  how  many  weeks  in 
23044  days  ?  Ajis,  3292  weeks. 

21.  If  5  bushels  of  wheat  make  1  barrel  of  flour,  how 
many  barrels  of  flour  can  be  made  from  314670  bushels  ? 

Ans.  62934  barrels. 

22.  By  reading  9  pages  a  day,  how  many  days  will  be 
required  to  read  a  book  through  which  contains  1161 
pages?  A71S.  129  days. 

23.  At  4  dollars  a  yard,  how  many  yards  of  broadcloth 
can  be  bought  for  1372  dollars  ?  Ans.  343  yards. 

24.  If  a  stage  goes  at  the  rate  of  8  miles  an  hour,  how 
long  will  it  be  in  going  1560  miles  ?      Ans.  195  hours. 

25.  There  are  3  feet  in  1  yard ;  how  many  yards  in 
206175  feet?  Ans.  6Sn5  yards. 

26.  Five  partners  share  equally  the  loss  of  a  ship  and 
cargo,  valued  at  760315  dollars ;  what  is  each  one's 
share  of  the  loss  ?  Ans.  152063  dollars. 

27.  If  a  township  of  64000  acres  be  divided  equally 
among  8  persons,  how  many  acres  will  each  receive  ? 

Ans.  8000  acres. 

28.  A  miller  wishes  to  put  36312  bushels  of  grain  into 
6  bins  of  equal  size ;  how  many  bushels  must  each  bin 
contain?  Ans.  6052  bushels. 

29.  How  many  steps  of  3  feet  each  would  a  man  take 
in  v>^alking  a  mile,  or  5280  feet?  Ans.  1760  steps. 

30.  A  gentleman  left  his  estate,  worth  36105  dollars,  to 
be  shared  equally  by  his  wife  and  4  children  ;  what  did 
each  receive?  Ans.  7221  dollars. 


62  simple   numbers. 

Case  IL 

70.  When  the  divisor  consists  of  two  or  more  fig- 
ures. 

To  illustrate  more  clearly  the  method  of  operation,  we  will  first  take  an  ex- 
ample usually  performed  by  Short  Division. 

1.  How  many  times  is  4  contained  in  1504  ? 

OPERATION.  Analysts.    First.  Seek  how  many  times  the 

4)  1504  (  376         divisor,  4,  is  contained  in   15,  the  fi^st  partial 

12  dividend,  which  we  find  to  be  3  times,  and  a 

„„  remainder.     Place  this  quotient  figure  at  the 

(jQ  right  hand  of  the  dividend,  with  a  line  between 

them.     Second.  To  find  the  remainder,  multiply 

24  the  divisor,  4,  by  this  quotient  figure,  3,  and 

24  place  the  product,  12,  under  the  figures  divided. 

'  Subtracting  the  product  from  the  figures  di- 
vided, there  is  a  remainder  of  3.  Third.  Bringing  down  the  next 
figure  of  the  dividend  to  the  right  hand  of  the  remainder,  we  have 
30,  the  second  partial  dividend.  Then  4  is  contained  in  30,  7  times 
and  a  remainder.  Placing  the  7  at  the  right  hand  of  the  last  quo- 
tient figure,  and  multiplying  the  divisor  by  it,  we  place  the  product, 
28,  under  the  figures  last  divided,  and  subtract  as  before.  To  the 
remainder,  2,  bring  down  the  next  figure,  4,  of  the  given  dividend, 
and  we  have  24  for  the  third  partial  dividend.  Then  4  is  contained 
in  24,  6  times.  Multiplying  and  subtracting  as  before,  nothing  re- 
mains, and  we  have  for  the  entire  quotient  370. 

When  the  whole  proces=!  of  division  is  written  out  as  above,  the  operation  if 
termed  Long  Division.    The  princi^,  however,  is  the  same  as  Short  Division. 

Solve  the  following  examples  by  Long  Division  : 

2.  Divide  4672  by  8.  Ans.      584. 

3.  Divide  97636  by  7.  Ans.  13948. 

4.  Divide  37863  by  9.  A7is,    4207. 

5.  Divide  394064  by  11.  Ans.  35824. 


DIVISION.  63 

6.  How  many  times  is  23  contained  in  17158  ? 

OPERATION.  Analysis.    As  23  is  not  contained  in  the 

23  )  17158  (  746     first  two  figures  of  the  dividend,  find  how 

1  gi  many  times  it  is  contained  in  171,  as  the  first 

~  partial  dividend.     23  is  contained  in  171,  7 

1^^  times,  which  place  in  the  quotient  on  the 

92  right  of  the  dividend.     Then  multiply  the 

-joQ  divisor  23,  by   the   quotient   figure   7,  and 

subtract  the  product  161,  from  the  part  of 

.  the  dividend  used,  and  we  have  a  remainder 

of  10.      To  this  remainder  bring  down  the 

next  figure  of  the  dividend,  making   105  for  the  second  partial 

dividend.     Then,  23  is  contained  in  105,  4  times,  which  place  in 

the  quotient.     Multiplying  and  subtracting  as  before,  we  have  a 

remainder  of  13,  to  w^iich  bring  down   the   next  figure   of  the 

dividend,  making  138  for  the  third  partial  dividend.   23  is  contained 

in  138,  6  times  ;    multiplying  and  subtracting  as  before,  nothing 

remains,  and  we  have  for  the  entire  quotient,  740. 

From  the  preceding  illustrations  we  derive  the  following 
general 

EuLE.  I.  Write  the  divisor  at  tlie  left  of  the  dividend, 
as  in  short  division. 

II.  Divide  the  least  number  of  tJie  left  hand  figures  in 
the  dividend  that  will  contain  the  divisor  one  or  more  times, 
a7id  place  the  quotient  at  the  right  of  the  dividend,  tvith  a 
line  hetiveen  them. 

HI.  Multiply  the  divisor  iy  this  quotient  figure,  suUract 
the  product  from  the  partial  dividend  used,  and  to  the 
remainder  bring  doiun  the  next  figure  of  the  dividend. 

TV.  Divide  as  before,  until  all  the  figures  of  the  dividend 
"have  been  brought  doivn  and  divided. 

V,  If  ajiy  partial  dividend  will  not  contain  the  divisor, 


64  SIMPLE     NUMBERS, 

place  a  cipher  in  the  quotient,  and  bring  down  the  next 
figure  of  the  dividend,  and  divide  as  before. 

VL  If  there  be  a  remainder  after  dividing  all  the  figures 
of  the  dividend,  it  must  be  written  in  the  quotient,  with  the 
divisor  underneath. 

1.  If  any  remainder  be  equal  to^  or  greater  than  the  divisor,  the  quotient 
figure  is  too  small,  and  must  be  increased. 

2.  If  the  product  of  the  divisor  by  the  quotient  figure  be  greater  than  the  par- 
tial dividend,  the  quotient  figure  is  too  large,  and  must  be  diminished. 

Prooe.     The  same  as  in  short  division. 

71.  The  operations  in  long  division  consists  of  five 
principal  steps,  viz. : — 

1st.  Write  down  the  numbers. 

2d.  Find  how  many  times. 

3d.   Multiply. 

4th.  Subtract. 

5th.  Bring  down  another  figure. 

Examples  for  Practice. 
7.  Find  how  many  times  18  is  contained  in  36838. 


OPERATION. 

PROOF 

Dividend.    Quotient. 

Divisor.   18 )  36838  ( 2046i-| 

2046 

Quotient. 

86 

18 

Divisor. 

83 

16368 

72 

2046 

118 

36828 

108 

10 

Remainder. 

1 0    Remainder. 

36838 

Dividend. 

DIVISI02S". 


65 


8.  Divide  79638  by  36. 

OPERATION. 

36)79638(2212^ 

72_ 

76 

72 

43 
36 


78 
72 

6    Eem. 


9.  Divide  93975  by  84. 

OPERATION. 

84)  93975  (1118f}. 
84 

99 
84_ 

157 
84 


735 
672 


63    Kern. 


10.  Divide  408722  by  136.      11.  Divide  104762  by  109. 


OPERATION. 

136  )  408722  (  3005 
408 

722 
680 


42    Rem. 


12.  Divide 

13.  Divide 

14.  Di^dde 

15.  Divide 

16.  Di\'ide 

17.  Divide 

18.  Di^dde 


178464  by  16, 
15341  by  29. 
463554  by  39. 
1299123  by  17. 
161700  by  15. 
47653  by''24. 
765431  by  42. 


OPERATION. 

109  )  104762  ( 961 

981 

666 

654 

122 

109 

13  Rem. 

Ans.   11154. 

A71S.      529. 

Ans.   11886. 

Ans.   76419. 

Ans.   10780. 

>-^/y 

66 


SIMPLE    NUMBEBS. 


19. 

20. 
21. 

22. 
23. 
24. 
25. 

26. 

27. 
28. 

30. 
31. 
32. 
33. 
34. 
35. 
^36. 
37. 
38. 
39. 
40. 
41. 
42. 

43. 

44. 

45. 
what 

46. 
what 


Di\icle  6783  by  15. 
Divide  7831  by  18. 
Divide  9767  by  22. 
Divide  7654  by  24. 
Divide  767500  by  23. 
Divide  2507G5  by  34. 
Divide  5571489  by  43. 
Divide  153598  by  29. 
Divide  301147  by  63. 
Divide  40231  by  75. 
Divide  52761878  by  126. 
Divide  92550  by  25. 
Divide  7461300  by  95. 
Di^dde  1193288  by  45. 
Divide  5973467  by  243. 
Divide  69372168  by  342. 
Divide  863256  by  736. 
Divide  1893312  by  912. 
Divide  833382  by  207. 
Divide  52847241  by  607. 
Di^-ide  13699840  by  342. 
Divide  946656  by  1038. 
Divide  46447786  by  1234. 
Divide  28101418481 

by  1107.  25385201 

Divide  48288058  by  3094.  15607 

Divide  47254149  by  4674.  10110 

A  man  bought  114  acres  of  land  for  4104  dollars; 
was  the  average  price  per  acre  ?    Ans.  36  dollars. 
Nine  thousand  dollars  was  paid  to  75  operatives ; 
did  each  receive  ?  Ans.  120  dollars. 


Quotients. 
452 
435 
443 

318 
33369 

7375 

129569 

5296 

4780 

536 

418745 

3702 

78540 

26517 

24582 

S02842  ■ 

1172 

2076' 

4026 

87063 

40058 

912 

37640 


Rem. 

3. 

1. 

21. 
22. 
13. 
15. 
22. 
14. 

7. 
31. 

8. 


23. 

41. 
204. 
664. 


4. 


26. 


974. 


9. 


DIVISIOIS".  67 

47.  There  are  24  hours  in  a  day ;  how  many  days  in 
11424  hours  ?  _  Aits.  476  days. 

48.  In  one  hogshead  are  63  gallons ;  how  many  hogs- 
heads in  6615  gallons  ?  Ans.  105  hogsheads. 

49.  If  a  man  travel  48  miles  a  day,  how  long  mil  it 
take  him  to  travel  1296  miles?  Ans.  27  days. 

50.  If  a  person  can  count  8677  in  an  hour,  how  long 
will  it  take  him  to  count  38369694?    Ans.  4422  hours. 

51.  If  it  cost  5987520  dollars  to  construct  a  railroad 
676  miles  long,  what  will  be  the  average  cost  per  mile  ? 

Ans.  10395  dollars. 

52.  The  Memphis  and  Charleston  railroad  is  287  miles 
in  length,  and  cost  5572470  dollars  ;  what  \\'^ti  the  aver- 
age cost  per  mile?  Ans.  1941 6-^\  dollars. 

53.  A  garrison  consumed  1712  barrels  of  flour  in  107 
days  ;  how  much  was  that  per  day?      A7is.  16  barrels. 

54.  How  long  would  it  take  a  vessel  to  sail  from  New 
York  to  China,  allowing  the  distance  to  be  9072  miles,  and 
the  ship  to  sail  144  miles  a  day  ?  A  ns.  63  days. 

/ 65.  How  long  could  27  men  subsist  on  a  stock  of  provi- 
sion that  would  last  1  man  3456  days?    Ans.  128  days. 
^  56.  A  drover  received  10362  dollars  for  314  head  of 
cattle  ;  what  was  their  average  value  per  head  ? 

Ans.  33  dollars. 

57.  If  42864  pounds  of  cotton  be  packed  in  94  bales,  what 
is  the  average  weight  of  each  bale  ?     A7is.  456  pounds. 

58.  If  a  field  containing  42  acres  produce  1659  bushels 
of  wheat,  what  will  be  the  number  of  bushels  per  acre  ? 

Ans.  39|-J-  bushels. 

59.  In  what  time  will  a  reservoir  containing  109440 
gallons,  be  emptied  by  a  pump  discharging  608  gallons 
per  hour?     '  Ans.  180  hours. 


68  SIMPLE     NUMBERS. 

COKTRACTI0KS. 

Case  I. 
^2.  When  the  divisor  is  10,  100,  1090,  etc. 

1.  Divide  374  by  10. 

OPERATION.  Analysis.    Since  we  have  shown, 

llO  ")  37  4  *^^^*  *^  remove  a  figure  one  place  to- 

ward  the  left  by  annexing  a  cipher 

Quotient,     67         4  Rem.       increases  its  value  tenfold,  or  multi- 
or,  373*Q-,  Ans,  plies  it  by  10,  so,  on  the  contrary,  by 

cutting  ofl'  or  taking  away  the  right 
hand  figure  of  a  number,  each  of  the  other  figures  is  removed  one 
place  toward  the  right,  and,  consequently,  the  value  of  each  ia 
diminished  tenfold,  or  divided  by  10. 

For  similar  reasons,  if  we  cut  ofl  hvo  figures,  we  divide 
by  100,  if  three^  we  divide  by  1000,  and  so  on.     Hence  the 

EuLE.  From  the  right  hand  of  the  dividend  cut  off  as 
many  figures  as  there  are  ciphers  in  the  divisor.  Under 
the  figures  so  cut  off  place  the  divisor,  and  the  whole  will 
form  the  quotiejit. 

Examples  for  Practice. 

2.  Divide  13705  by  100.' 

3.  Divide  50670  by  100. 

4.  Divide  320762  by  1 000. 

5.  Divide  14030731  by  10000. 

6.  Divide  9021300640  by  100000. 

7.  A  man  sold  100  acres  of  land  for  3725  dollars  ;  what 
did  he  receive  an  acre?  Ans.  37y^xrg-  dollars. 

8.  Bought  1000  barrels  of  flour  for  6080  dollars  ;  what 
did  it  cost  me  a  barrel  ?  Aiis.  Qj^iy  dollars. 

9.  Paid  12560  dollars  for  10000  bushels  of  wheat ;  what 
was  the  cost  per  bushel  ?  Ans.  l^i%V  dollars. 


Dtients. 

Rem's. 

137 

5. 

606 

70. 

320 

762. 

1403 

731. 

90213 

640. 

DIYISIOIT.  69 

Case  II. 

73.  When  there  are  ciphers  on  the  right  hand  of 
the  divisor. 

1.  Divide  437661  by  800. 

OPERATION.  Analysis.     In  tLis  example  we  resolve 

8|00  )  4376161  800  into  the  factors  8  and  100,  and  divide 

.  ^  first  by  100,  by  cutting  oflf  two  right  hand 

figures  of  the  dividend,  and  we  have  a 

quotient  of  4376,  and  a  remainder  of  61.     We  next  divide  by  8,  and 

obtain  547  for  a  quotient ;  and  the  entire  quotient  is  547 /q^^. 

2.  Divide  34716  by  900. 

OPERATION,  Analysis.     Dividing  as  in  the  last 

9100)  347116  example,  we  have  a  quotient  of  38, 

~  ,.  ^  „  and  a  remainder  of  5  after  dividing 

Quotient,    38  -  -  -  0l6  Rem.     ^^  ^^  ^^^.^^  ^^  p^^^^  ^^  ^^^  ^.^^^^^^ 

or,  38|-Qf ,  A7IS.        f,^i  off  from  the  dividend,  making  a 
true  remainder  of  516,  and  the  entire  quotient  38|^. 

Rule.  I.  Cut  off  the  ciphers  from  the  right  of  the  divi- 
sor, and  the  same  number  of  figures  from  the  rigid  of  the 
divide7id. 

II.  Divide  the  remaining  figures  of  the  dividend  hy  the 
remainiifg  figures  of  the  divisor,  and  the  result  will  he  the 
quotient.  If  there  he  a  remainder  after  this  division,  pre- 
fix it  to  the  figures  cut  off  from  the  dividend,  and  this 
will  form  the  true  remainder. 

Examples  for  Practice. 

Quotients.  Rem's. 

8.  Divide  46820  by  400.  117  20. 

4.  Divide  130627  by  800.  163  227. 

5.  Divide  76173  by  320.  238  13. 

6.  Divide  378000  by  1200.       -  315 


VO  SIMPLE     NUMBEKS. 

7.  Divide  674321  by  11200.  60   2321. 

8.  Divide  64613214  by  4000.       16153   1214. 

9.  Divide  146200  by  430.  340 

10.  Divide  7380964  by  23000.        320  20964. 

11.  Divide  58677000  by  1800.       32598  ^   600. 

Examples  ik  the  Peecedikg  Rules. 

1.  A  speculator  bought  at  different  times  320  acres,  175 
acres,  87  acres,  and  32  acres  of  land,  and  afterward  sold  467 
acres  ;  how  many  acres  had  he  left  ?      A7is.  147  acres. 

2.  Two  men  travel  in  opposite  directions  ;  one  travels 
31  miles  a  day,  the  other  43  miles  a  day ;  how  far  apart 
will  they  be  in  12  days  ?  Ans.  888  miles. 

3.  A  tob^jC-3onist  has  6324  pounds  of  tobacco,  which  he 
wishes  uO  piwk  in  boxes  containing  62  pounds  each  ;  how 
many  boxes  must  he  procure  to  contain  it  ?    Aits.  102. 

4.  A  farmer  sold  15  tons  of  hay  at  9  dollars  a  ton,  and 
25  cords  of  wood  at  4  dollars  a  cord,  and  wished  to  divide 
the  amount  equally  among  5  creditors  ;  what  would  each 
receive?  Atis.  47  dollars. 

5.  If  you  deposit  216  cents  each  week  in  a  savings  bank, 
and  take  out  89  cents  a  week,  how  many  cents  will  you 
have  in  bank  at  the  end  of  36  weeks  ?  Ans.  4572  cents. 

6.  The  product  of  two  numbers  is  8928,  and  one  of  the 
numbers  is  72  ;  what  is  the  other  number  ?   Ans.  124. 

7.  The  dividend  is  7280,  and  the  quotient  is  208  ;  what 
is  the  divisor  ?  A7is.  35. 

8.  What  is  the  remainder  after  di\riding  876437  by 
16900  ?  Ans.  14537. 

9.  A  man  sold  6  horses  at  125  dollars  each,  25  head  of 
cattle  at  30  dollars  each,  and  with  the  proceeds  bought 
land  at  25  dollars  an  acre  ;  how  many  acres  did  he  buy? 

A?is.  60  acres. 


PROMISCUOUS     EXAMPLES.  71 

10.  If  a  mechanic  receives  784  dollars  a  year  for  labor, 
and  his  expenses  are  426  dollars  a  year,  how  much  can  he 
save  in  6  years?  Ans.  2148  dollars. 

11.  A  farmer  sold  40  bushels  of  wheat  at  2  dollars  a 
bushel,  and  16  cords  of  wood  at  3  dollars  a  cord.  He  re- 
ceived 15  yards  of  cloth  at  4  dollars  a  yard,  and  the  re- 
mainder in  money ;  how  much  money  did  he  receive  ? 

A71S.  68  dollars. 

12.  Hovf  many  pounds  of  cheese,  worth  10  cents  a  pound, 
can  be  bought  for  22  pounds  of  butter,  worth  15  cents  a 
pound  ?  A71S.  33  pounds. 

13.  If  56  yards  of  cloth  cost  336  dollars,  how  much  will 
12  yards  cost  at  the  same  rate  ?  Ans.  72  dollars. 

14.  If  100  barrels  of  flour  cost  600  dollars,  Vvhat  will 
350  barrels  cost  at  the  same  rate  ?     Ans.  2100  dollars. 

15.  How  long  can  60  men  subsist  on  an  amount  of  food 
that  will  last  1  man  7620  days  ?  Ajis.  127  days. 

16.  If  I  buy  225  barrels  of  flour  for  1125  dollars,  and 
sell  the  same  for  1800  dollars,  how  much  do  I  gain  on 
each  barrel  ?  Ans.  3  dollars. 

17.  A  man  sold  his  house  and  lot  for  5670  dollars,  and 
took  his  pay  in  bank  stock  at  90  dollars  a  share  ;  how 
many  shares  did  he  receive  ?  Ans.  63  shares. 

18.  How  many  pounds  of  tea,  worth  75  cents  a  pound, 
ought  a  man  to  receive  in  exchange  for  27  bushels  of  oats, 
worth  50  cents  a  bushel?  A^is.  18  pounds. 

19.  The  quotient  of  one  number  divided  by  another  is 
40,  the  divisor  is  364,  and  the  remainder  120  ;  what  is 
the  dividend  ?  A7zs.  14680. 

20.  How  many  tons  of  hay,  at  12  dollars  a  ton,  must  be 
given  for  21  cows  at  24  dollars  apiece  ?     A?is.  42  tons. 


72  SIMPLE     NUMBERS. 

21.  Bought  150  barrels  of  flour  for  1050  dollars,  and 
sold  107  barrels  of  it  at  9  dollars  a  barrel,  and  the  re- 
mainder at  7  dollars  a  barrel';  did  I  gain  or  lose,  and 
how  much?  A^is.  Gained  214  dollars. 

22.  A  mechanic  earns  45  dollars  a  month,  and  his 
necessary  expenses  are  27  dollars  a  month  ;  how  long  will 
it  take  him  to  pay  for  a  farm  of  50  acres,  at  27  dollars  an 
acre  ?  Ans.  75  months. 

23.  How  many  barrels  of  flour,  at  7  dollars  a  barrel,  will 
pay  for  30  tons  of  coal,  at  4  dollars  a  ton,  and  44  cords  of 
wood,  at  3  dollars  a  cord?  Ans.  36  barrels. 

Peoblems  ii^  Simple  Integral  Numbers. 

74.  The  four  operations  that  have  now  been  consid- 
ered, viz..  Addition,  Subtraction,  Multiplication,  and 
Division,  are  all  the  operations  that  can  be  performed 
uj)on  numbers,  and  hence  they  are  called  the  Funda- 
menial  Rides. 

75.  In  all  cases,  the  numbers  operated  upon,  and  the 
results  obtained,  sustain  to  each  other  the  relation  of  a 
whole  to  its  parts.     Thus, 

I.  In  Addition,  the  numbers  added  are  the  parts,  and 
the  sum  or  amount  is  the  whole. 

II.  In  Subtraction,  the  subtrahend  and  remainder  are 
the  parts,  and  the  minuend  is  the  whole. 

III.  In  Multiplication y  the  multiplicand  denotes  the 
value  of  one  part,  the  multiplier  the  number  of  parts,  and 
the  product  the  total  value  of  the  whole  number  of  parts. 

TV.  In  Division,  the  dividend  denotes  the  total  value  of 
the  whole  number  of  parts,  the  divisor  the  value  of  one  part, 
and  the  quotient  the  number  of  parts  ;  or  the  divisor  the 
number  of  parts,  and  the  quotient  the  value  of  one  part. 


PBOBLEMS.  73 

76.  Let  tlie  pupil  be  required  to  illustrate  the  follow- 
ing problems  by  original  examples  : 

Problem  1.   Griven,  several  numbers,  to  find  their  sum. 

Prob.  2.  Given,  the  sum  of  several  numbers  and  all  of 
them  but  one,  to  find  that  one. 

Prob.  3.  Given,  two  numbers,  to  find  their  difference. 

Prob.  4.  Given,  the  minuend  and  subtrahend,  to  find 
the  remainder. 

Prob.  5.  Given,  the  minuend  and  remainder,  to  find 
the  subtrahend. 

Prob.  6.  Given,  the  subtrahend  and  remainder,  to  find 
the  minuend. 

Prob.  7.  Given,  two  or  more  numbers,  to  find  their 
product. 

Prob.  8.  Given,  the  multiplicand  and  multiplier,  to 
find  the  product. 

Prob.  9.  Given,  the  product  and  multiplicand,  to  find 
the  multiplier. 

Prob.  10.  Given,  the  jDroduct  and  multiplier,  to  find 
the  multiplicand. 

Prob.  11.  Given,  two  numbers,  to  find  their  quotients. 

Prob.  12.  Given,  the  divisor  and  dividend,  to  find  the 
quotient. 

Prob.  13.  Given,  the  divisor  and  quotient,  to  find  the 
di^ddend. 

Prob.  14.  Given,  the  dividend  and  quotient,  to  find  the 
divisor. 

Prob.  15.  Given,  the  divisor,  quotient,  and  remainder, 
to  find  the  dividend. 

Prob.  16.  Given,  the  dividend,  quotient,  and  remain- 
der, to  find  the  divisor. 

4 


74  FRACTIOKS. 


FEACTIONS. 

Definitions,  Notation,  and  [N'umeeation. 

77.  If  a  unit  be  divided  into  2  equal  parts,  one  of  the 
parts  is  called  07ie  half. 

If  a  unit  be  divided  into  3  equal  parts,  one  of  the  parts 
is  called  oiie  thirds  two  of  the  parts  tivo  thirds. 

If  a  unit  be  divided  into  4  equal  parts,  one  of  the  parts 
is  called  one  fourth,  two  of  the  parts  two  fourths,  three  of 
the  parts  three  fourths. 

If  a  unit  be  divided  into  5  equal  parts,  one  of  the  parts 
is  called  07ie  fifth,  two  of  the  parts  two  fifths,  three  of  the 
parts  three  fifths,  etc. 

And  since  ojie  half,  one  third,  one  fourth,  and  all  other 
equal  i^arts  of  an  integer  or  -whole  thing,  are  each  in  them- 
selves entire  and  complete,  the  ])arts  of  a  unit  thus  used 
are  called  fractional  units  ;  and  the  numbers  formed  from 
them,  fractional  7iuwl)ers.     Hence, 

78.  A  Fractional  Unit  is  one  of  the  equal  parts  of  an 
integral  unit. 

79.  A  Fraction  is  a  fractional  unit,  or  a  collection  of 
fractional  units. 

80.  Fractional  units  take  their  name,  and  their  value, 
from  the  numler  of  parts  into  which  the  integral  unit  is 
divided.  Thus,  if  we  divide  an  orange  into  2  equal  parts, 
the  parts  are  called  halves  ;  if  into  3  equal  parts,  thirds  ; 
if  into  4  equal  parts,  fourths,  etc.;  and  each  third  is  less  in 
value  than  each  half,  and  each  fourth  less  than  each  third  ; 
and  the  greater  the  number  of  parts,  the  less  their  value. 


NOTATIOif     AliTD     KUMERATION.  75 

The  parts  of  a  fraction  are  expressed  by  figures  ;  thus. 


One  half    is  written 

i 

One  fifth      is  written 

\ 

One  third          " 

i 

Two  fifths 

f 

Two  thirds        *^ 

o 

One  seventh 

\ 

One  fourth        ^' 

i 

Three  eighths 

t 

Two  fourths      '^ 

2 

■4 

Five  ninths 

1 

Three  fourths    ^^ 

3 

4 

Eight  tenths 

A 

To  write  a  fraction,  therefore,  two  integers  are  required, 
one  written  above  the  other  with  a  Hne  between  them. 

81.  The  Denominator  of  a  fraction  is  the  number 
below  the  line.  It  shows  into  how  many  parts  the  inte- 
ger or  unit  is  di^'ided,  and  determines  the  value  of  the 
fractional  unit, 

82.  The  I^unierator  is  the  number  above  the  line. 
It  numbers  the  fractional  units,  and  shows  how  many 
are  taken. 

83.  Thus,  if  one  dollar  be  divided  into  '_  equal  parts^ 
the  parts  are  called /o?<r///5,  the  fractional  i:iiit  being  one 
fourth,  and  three  of  these  parts  are  called  iliree  fourths 
of  a  dollar,  and  may  be  written 

3  the  number  of  parts  ox  fractional  units  taken. 

4  the  number  of  parts  ox  fractional  units  into  which  the  dollar  is  divided. 

84.  The  Terms  of  a  fraction  are  the  numerator  and 
denominator,  taken  together. 

S^,  Fractions  inclicate  division,  the  numerator  answer- 
ing to  the  dividend,  and  the  denominator  to  the  divisor. 
Hence^ 

SQ.  The  Value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator. 

Thus,  the  quotient  of  4  divided  by  5  is  f,  or  |  expresses 
the  quotient  of  which  i  |  g  [iJI  ZS^' 


76  FEACTIOI^S. 

1.  What  is  1  half  of  8  ? 

Analysis.  It  is  tlie  quotient  of  8  divided  by  2,  wliicli  is  4 ;  or, 
it  is  a  number  which,  taken  2  times,  will  make  8,  which  is  4. 
Therefore,  4  is  1  half  of  8.  . 

2.  What  is  2  thirds  of  9  ? 

Analysis.  Since  1  third  of  9  is  3,  2  thirds  of  9  is  2  times  3, 
which  is  6.     Therefore,  2  thirds  of  9  is  6. 

Hence,  to  obtain  one  half,  one  third,  one  fourth,  or  any 
fractional  ipart  of  a  number,  di^ade  that  number  by  the 
denominator  of  tlie  fraction  expressing  the  jmrts  ;  and  to 
obtain  any  given  number  of  such  parts,  multi'ply  that 
part  hy  the  number  of  parts  expressed  hy  the  numerator 
of  tlie  same  fraction. 

3.  What  is  1  fourth  of  12  ?  3  fourths  of  12  ? 

4.  What  is  1  fifth  of  20  ?  3  fifths  ?  4  fifths  ? 

5.  What  is  1  eighth  of  40  ?  3  eighths  ?  5  eighths  ? 

6.  What  is  2  sevenths  of  21  ?  5  sevenths  of  35  ?  6  sev- 
enths of  49  ? 

7.  What  is  1  ninth  of  63?  2  ninths  of  27  ?  4  ninths  of 
36  ?  5  ninths  of  45  ?  7  ninths  of  81  ? 

8.  What  is  1  twelfth  of  48  ?  5  twelfths  ?  7  twelfths  ? 

9.  If  a  pound  of  coffee  cost  15  cents,  vviiat  will  1  third 
of  a  pound  cost  ?  2  thirds  ? 

10.  A  farmer  having  60  sheep,  sold  1  fifth  of  them  to 
one  man,  and  3  fifths  to  another  X  how  many  did  he  sell 
to  both  ? 

11.  A  boy  having  48  cents,  spent  3  eighths  of  them; 
how  many  had  he  left  ? 

12.  Paid  108  dollars  for  a  horse,  and  9  twelfths  as  much 
for  a  carriage  ;  what  did  the  carriage  cost  ? 

13.  William  had  120  pennies,  and  James  had  7  tentlui' 
as  many  ;  how  many  had  James  ? 


NOTATION     AND     NUMERATION.  77 

87.  It  is  often  reqviired  to  express  by  a  fraction,  what 
part  one  number  is  of  another  number. 

1.  What  part  of  5  is  3  ? 

Analysis.    Since  1  is  1  fifth  of  5,  3  is  3  times  1  fifth  of  5,  or  3 
fifths  of  5.     Therefore,  3  is  3  fifths  of  5. 

The  number  preceded  by  the  word  of  is  generally  made  the  denominator  or 
divisor,  and  the  other  number  called  the  part,  the  numerator  or  dividend. 

2.  What  part  of  6  is  3  ?  4  ?  5  ?  1  ? 

3.  What  part  of  9  is  2?  3  ?  5  ?  G  ?  1  ?  4? 

4.  What  part  of  10  is  7  ?  6  ?  3  ?  1  ?  9  ?  8  ?  4  ? 

5.  What  part  of  12  is  3?  5?  6?  8?  9?  7?  10?  11? 

6.  What  part  of  14  is  5  ?  7  ?  9  ?  3  ?  6  ?  11  ?  8  ?  15  ? 

7.  What  part  of  15  bashels  is  3  bushels  ?   7  bushels  ?  9 
bushels?  11  bushels? 

8.  What  part  of  18  dollars  is  7  dollars?    5  dollars?    9 
dollars?  17  dollars?  . 

0.  If  6  oranges  cost  30  cents,  what  part  of  30  cents  will 
1  orange  cost  ?  2  oranges  ?  3  oranges  ?  5  oranges  ? 


Examples  in  Weiting  and  Reading  Feactions. 
Express  the  following  fractions  by  figures  : 

1.  9  Uoelftlis,  Ans.  -f^. 

2.  'EleYen  Jiftee}ifhs.  Ans.  -f^. 

3.  Twentj-fouY forty-ninfJis.  Ans.  ff. 

4.  Forty-four  sixty -ninths.  Ans.  |4' 

5.  One  hundred  twenty /o^^r  hundred  fiftieths. 

Read  the  follomng  fractions  : 

6.  A,  H,  II,  -M--  m,  ^>  w,  m- 

7.  If  the  fractional  unit  is  28,   express   9   fractional 
nnits  ;  16  ;  17  ;  22  ;  27. 

8.  If  the  fractional  unit  is  96,  express  27  fractional 
loiits ;  42  ;  75. 


78  FRACTIOIS'S. 

88.  Fractions  are  distingnislied  as  Proper  and  Improper, 
A  Proper  Fraction  is  one  whose  numerator  is  less  than 

its  denominator  ;  its  yalue  is  less  than  the  unit,  1.    Thus, 
A?  TS"?  TO  J  if  ^^^  proper  fractions. 

An  Improper  Fraction  is  one  whose  numerator  equals 
or  exceeds  its  denominator;  its  value  is  never  less  than  the 
unit,  1.     Thus,  -f.  |,  J^,  -^,  \^,  ^\'^are  improper  fractions- 

89.  A  Mixed  I^iimber  is  a  number  expressed  by  an 
integer  and  a  fraction  ;  thus,  4^,  17i-|?  ^fV  ^^^  mixed 
numbers. 

EEDUCTION.' 

90.  The  Reduction  of  a  fraction  is  the  process  of 
changing  its  form  without  altering  its  value. 

Case  I. 

91.  To  reduce  fractions  to  their,  lowest  terms. 

A  fraction  is  in  its  loivest  terms  when  no  number  greater 
than  1  will  exactly  divide  both  numerator  and  denomi- 
nator without  a  remainder. 

1.  Keduce  f  to  its  lov/est  terms. 

Ajstalysis.  It  is  plain,  that  the  numerator  2,  and  the  denom- 
inator 4,  are  both  divisible  by  2,  without  remainders ;  hence 

4-^2~2* 
The  terms  thus  obtained,  viz.,  1,  the  numerator,  and  2,  the  de- 
nominator, are  not  both  divisible  by  any  number  greater  than  1 ,  and 
therefore  are  the  smallest  terms  by  which  the  value  of  f  can  be 
expressed. 

2.  Reduce  f  to  its  lowest  terms. 

3.  Reduce  -f^  to  its  lowest  terms. 

4.  Reduce  f  to  its  lowest  terms. 

5.  Reduce  -/-g-  to  its  lowest  terms. 

6.  Reduce  ^g-  to  its  lowest  terms. 


REDUCTION-.  79 

7.  Reduce  ^  to  its  lowest  terms. 

OPERATION.  Analysis.    Dividing  both  terms 

9\48 24.   9\2_i — 12.  of  a  fraction  by  the  same  number 

o\t2 4     4)1'^  ^^^           alter  the  value   ot   the 

'  ^^T7o       a       4  fraction  or  quotient  ;    hence,   we 

r\r^      1  9    4    4^ 4.       A  7)Q  ^ 

01,  i^  ;  6  0  —  5^  ^^^^^'  divide  both  terms  of  ||  by  2,  and 

obtain  |^- ;  dividing  both  terms  of  this  fraction  by  2,  we  have  jf  as 
the  result ;  finally,  dividing  the  terms  of  this  fraction  by  3,  we  have 
-4  and  as  no  number  greater  than  1  will  divide  the  terms  of  this 

5  *  ^ 

fraction  "v\athout  a  remainder,  f  are  the  lowest  terms  in  which  the 
value  of  1^  can  be  expressed.  We  may  obtain  the  final  result 
more  readily,  by  dividing  the  terms  of  this  fraction  by  the  largest 
number  that  will  divide  both  without  a  remainder  ;  dividing  by  12, 
we  obtain  f ,  the  answer. 

Rule.  Divide  the  terms  of  the  fraction  hij  any  number 
greater  than  1,  that  will  divide  both  without  a  remainder^ 
and  the  quotients  obtained  in  the  same  manner,  until  no 
number  greater  than  1  loill  so  divide  them  ;  the  last  quo- 
tients will  be  the  loioest  terms  of  the  given  fraction. 

Examples  foe  Pkactice. 

8.  Reduce  -ff  to  its  lowest  terms.  Ans.  f. 

9.  Reduce  yW  to  its  lowest  terms.  Ans.  f. 

10.  Reduce  -^f^  to  its  lowest  terms.  Ans.  -|. 

11.  Reduce -J-rl  to  its  lowest  terms.  Ans.  f. 

12.  Reduce  |^f  to  its  lowest  terms.  Ans.  \^. 

13.  Reduce  f|f  to  its  lowest  terms.  Ans.  |ff. 

14.  Reduce  f^f  to  its  lowest  terms.  Ans.  f. 

15.  Reduce  -^-^  to  its  lowest  terms.  Ans.  \. 

16.  Reduce  |||-  to  its  lowest  terms.  Ans.  |f . 

17.  Reduce  -ff^  to  its  lowest  terms.  Ans.  \. 

18.  Reduce  \^\  to  its  lowest  terms.  Ans.  ^%. 

19.  Reduce  ffff  to  its  lowest  terms.  Ans.  -Hfj-. 


80  FKACTIOKS. 

Case  II. 

93.  To  change  an  improper  fraction  to  a  whole  or 
mixed  number. 

1 .  In  J/-  how  many  times  1  ? 

Analysis.  Since  1  equals  |,  ^-  equals  as  many  times  1,  as  f 
are  contained  times  in  Jj--,  which  are  3  times.  Therefore,  ^  are  3 
times  1,  or  3. 

2.  How  many  times  1  in  ^-  ?  in  ^  ?  in  ^■^-  ? 

3.  How  many  times  1  in  ^  ?  in  -^^  ?  in  -^  ? 

4.  How  many  times  1  in  -^-  ?  in  -f-§-  ?  in  ^  ? 
6.  How  many  times  1  in  -^-f-  ?  in  -f^  ?  iii  -fr  ^ 

When  the  denominator  is  not  an  exact  divisor  of  the  numerator,  the  resnlt 
Will  be  a  mixed  number. 

6.   In  ^-  how  many  times  1  ? 

OPERATION.  Analysis.      Since   1   equals  |,  -V-  equal 

♦  /  ^"  as  many  times  1  as  7  is  contained  times  in 

2f ,  Ans.      16,  which  is  2f  times. 

EuLE.  Divide  the  numerator  ly  the  denominator. 

Examples  foe  Pkactice. 

7.  In  J-|-^  how  many  times  1  ?  Ans.  244-. 

8.  In  ^-f^  of  a  year  how  many  years?  Ans.  19. 

9.  In  ^11^  of  a  pound  how  many  pounds  ?  Ans.  107. 

10.  In  f  ^1  of  a  mile  how  many  miles  ?  Ans.  G. 

11.  In  '^1%'^  of  a  rod  how  many  rods ?  Ans.  212|-. 

12.  In  "  f  ^  ^  of  a  dollar  how  many  dollars  ? 

13.  Reduce  ff  to  a  whole  number.  Ans.  6. 

14.  Reduce  -^  to  a  mixed  number.  Ans,  5-|. 

15.  Reduce  ^f^^  to  a  whole  number.  ^?zs.  18. 

16.  Reduce  ^41-^  to  a  mixed  number.  ./4?^5.  601- 

17.  Change -2-|A-§- to  a  mixed  number.  ^?2s.  67|. 

18.  Change  -^f^  to  a  whole  number.  Ans,  52. 


BEDUCTION.  81 

Case  III. 

93.  To  reduce  a  whole  or  mixed  number  to  an 
improper  fraction. 

1.  How  many  thirds  in  4  ? 

Analysis.  Since  in  1  there  are  3  thirds,  in  4  there  are  4  times  3 
thirds,  or  12  thirds.     Therefore,  there  are  ^  in  4. 

2.  How  many  fourths  in  2  ?  in  3  ?  in  5  ? 

3.  How  many  halves  in  5  ?  in  7  ?  in  8  ?  in  9  ? 

4.  How  many  sixths  in  3  ?  in  5  ?  in  7  ?  in  10  ? 

5.  How  many  tenths  in  4  ?  in  8  ?  in  9  ?  in  6  ? 

6.  How  mojij  fifths  in  2  whole  oranges  ?  in  4  ?  in  5  ? 

7.  How  many  eighths  in  4  whole  dollars  ?  in  5  ?  in  6  ? 

8.  In  3-|  dollars  how  many  eighths  of  a  dollar? 

OPERATION. 

gs  Analysis.     Since   in   1   dollar  there  are   8 

g  eighths,  in  3  dollars  there  are  3  times  8  eighths, 

—  or  24  eighths,  and  5  eighths  added,  make  -^#. 

24  +  5^=2/  ^       '  6  >  8 

EuLE.  Multiply  the  ichole  numher  hy  the  denominator 
of  the  fraction ;  to  the  product  add  the  numerator,  and 
under  the  result  tvrite  the  denominator. 

Examples  for  Practice. 

9.  Eeduce  6f  to  an  improper  fraction.         Ans.  ^■^-.■ 

10.  Eeduce  7|-  to  an  improper  fraction.  Ans.  ^. 

11.  Eeduce  15  to  a  fraction  whose  denominator  is  7. 

Ans.  ^K 

12.  Eeduce  120  to  ttvelfths.  Ans.  ^^. 

13.  In  242f  of  an  acre  how  many  thirds  of  an  acre? 

14.  In  75-|  hushels  how  many  eighths?         Ans.  ^^. 

15.  In  24  pounds  how  many  sixteenths  f      Ans.  ^^. 

16.  In  52  weeks  how  many  sevenths  9  Ans.  ^^-^. 

17.  Change  14^^  to  an  improper  fraction.  Ans.  %Y"- 


82  FEACTION-S. 

Case  IV. 

94.  To  reduce  two  or  more  fractions  to  a  common 
denominator. 

A  Common  Denominator  is  a  denominator  common 
to  two  or  more  fractions. 

Any  number  that  can  be  divided  by  each  of  the  denominators  of  the  given 
fractions,  may  be  taken  for  the  common  denominator. 

1.  Reduce  ^_  and  f  to  fractions  having  a  common  de- 
nominator. 

Analysis.  12  is  exactly  divisible  by  4  and  3,  and  may  therefore 
be  taken  for  a  common  denominator.  Since  in  1  there  are  ||,  in  \ 
of  1  there  must  be  l  of  {§,  or  y%  ;  and  in  f  of  1  there  must  be  |  of 
^f ,  or  Y^^,     Therefore  |-  and  |  are  equal  to  -^^  ^^d  tV 

2.  Reduce  f  and  |-  to  a  common  denominator. 

3.  Reduce  ^  and  -|  to  a  common  denominator. 

4.  Reduce  ^  and  -|  to  a  common  denominator. 

5.  Reduce  f  and  -|  to  a  common  denominator. 


OPEKATION. 

5x5  =  25 


Analysis.    Multiply  the  terms  of  the  first  frac- 
tion f,  by  the  denominator  5  of  the  second,  and 
6  X  5  =  30       the  terms  of  the  fraction  f,  by  the  denominator  6 
of  the  first.     This   must  reduce  each  fraction  to 
3  X  6  =  18       the  same  denominator  30,  for  each  new  denominator 
K  y  /^ OQ       will  be  the  product  of  the  given  denominators. 

Rule.    MuUiphj  loth  terms  of  each  fraction  by  the  de' 
nomi7iators  of  all  the  other  fractions. 

Mixed  numbers  must  first  be  reduced  to  improper  fractions. 

Examples  for  Practice. 

1.  Reduce  -f-  and  J  to  a  common  denominator. 

A71S.  if,  IJ. 


ADDITION.  83 

7.  Reduce  |-  and  -J  to  a  common  denominator. 

4^Q      3  2       3.5_ 

8.  Reduce  f  and  f  to  a  common  denominator. 

*  ^??o    14     i_s. 

9.  Reduce  -J-  and  ^^  to  a  common  denominator. 

10.  Reduce  |  and  -^  to  a  common  denominator. 

/J  4?  9     A  6.     4  0 

11.  Reduce  |-,  f ,  and  f  to  a  common  denominator. 

12.  Reduce  |-,  f ,  and  -f  to  a  common  denominator. 

J±iib.   Y^-^-,  jQ-^,  yYt- 

13.  Reduce  f,  -J-,  and  f  to  a  common  denominator. 

An^      1  ?2        40       1  50 
.o./«.>.    -o^fo,    24Q.    240* 

14.  Reduce  1^,  f ,  and  ^  to  a  common  denominator. 

/f  7?  <?      1  OS      54      5  6 

15.  Reduce  -{'q,  2J,  and  f  to  a  common  denominator. 

A'}')^     J-6  8.     540      2Q_Sl 
.ciicc.     2405    240?    24  0* 

;^6.  Reduce  -f^i  3|-,  f ,  and  f  to  a  common  denominator. 

A 01^      1  g  0      X2JLQ.      288      240 
.j^iLO.     §g  Q,      360?    360?    36  0* 

ADDITION. 

95.  The  denominator  of  a  fraction  determines  the  value 
of  the  fractional  unit ;  hence, 

I.  If  two  or  more  fractions  have  the  same  denominator, 
their  numerators  express  fractional  units  of  the  same  value. 

II.  If  two  or  more  fractions  have  different  denominators, 
their  numerators  express  fractional  units  of  different  values. 

And  since  units  of  the  same  value  onlv  can  be  united 
into  one  sum.  it  follows, 

III.  That  fractions  can  be  added  only  when  they  have 
the  same  fractional  unit  or  common  denominator. 


84  FRACTIONS. 

1.  What  is  the  sum  of  |,  |,  f ,  |  ? 

Analysis.  When  fractions  have  a  common  denominator,  their 
sum  is  found  by  adding  their  numerators,  and  placing  the  sum 
over  the  common  denominator.  Thus,  1  +3  +  4  +  2=10,  the  sum  of 
the  numerators  ;  placing  tiiis  sum  over  the  common  denominator  5, 
we  have  ^^-=2,  the  required  sum. 

2.  What  is  the  sum  of  ■^,  ■^,  und  ^? 
.   3.  What  is  the  sum  of  |,  f,  \,  and  -f? 

4.  What  is  the  sum  of  -J,  f ,  ^,  -|,  and  f  ? 

5.  A  boy  paid  f  of  a  dollar  for  a  pair  of  gloves,  f  of  a 
dollar  for  a  knife,  and  -J-  of  a  dollar  for  a  slate  ;  what  did 
he  pay  for  all  ? 

6.  A  father  distributed  some  money  among  his  children, 
as  folloYv^s  :  to  the  first  he  gave  -3^  of  a  dollar,  to  the  sec- 
ond -^,  to  the  third  y%,  to  the  fourth  y%,  and  to  the  fifth 
■^ ;  what  did  he  give  to  all  ? 

7.  What  is  the  sum  of  f  and  f  ? 

OPERATION.         "  Analysis,    As  the  given 

fj_2  —  21  I     8  — 35      AnQ  fractions  have  not  a  com- 

mon  denommator,  refiuce 
them  to  the  same  fractional  unit,  (94)  and  then  add  their  numer- 
ators, 27  +  8=35  ;  placing  the  sum  over  the  common  denominator 
36,  we  obtain  |f .     Hence  the  following 

Rule.  I.  When  the  given  fractions  have  the  same  de- 
nominator,  add  the  numerator Sy  and  under  the  sum  ivrite 
the  common  denominator. 

II.  When  they  have  not  the  same  denominator,  reduce 
th^m  to  a  common  denojninator,  and  then  add  as  Jjefore. 

If  the  amount  be  an  improper  fraction,  reduce  it  to  a  whole  or  a  mixed  number. 

Examples  tor  Peactice. 

8.  Wliat  is  the  sum  of  |  and  f?  Ans.  1-:^. 

9.  What  is  the  sum  of  ^  and  I?  Ans.  ||. 


\ 


ADDITION.  85 

10.  What  is  the  sum  of  |  and  |  ?  Ans.  |^. 

11.  Add  -f,  I,  and  |-  together.  A?is.  1-J. 

12.  Add  -f,  h,  and  |-  together.  Ans.  1^^. 

13.  Add  3^,  I,  |,  and  |  together.  Ans.  2-|. 

14.  Add  I,  i,  and  -J-^  together.  ^?i5.  23V 

15.  Add  f,  I,  f ,  and  |  together.  ^?Z5.  2-^-f-^. 

16.  What  is  the  sum  of  f,  f,  and  f  ?  ^?2s.  l-J^f^-. 

17.  What  is  the  sum  of  f ,  |-^  and  f  ?  ^;z5.  IfJ-. 

18.  What  is  the  sum  of  |,  -|,  and  -|?  Ans.  2^. 

To  add  mixed  numbers,  add  ike  fractions  and  integers 
separately,  and  then  add  their  sums. 

If  the  mixed  numbers  are  small,  tliey  may  be  reduced  to  improper  fractious, 
and  then  added  after  the  usual  method. 

19.  What  is  the  sum  of  14|,  21-J-,  and  9J? 


OPERATION. 
141==:14^«^ 


Analysts.    By   reducing  the  frac- 

^      -^■^■2  0"  tions  to  a  common  denominator,  and 

f)-|  j^ f)-|  1  0 

/c±2 — -^-^so  adding  them,  we   obtain  |§,  or  11^, 

9|-=:    9^  which  added  to  the  sum  of  the  inte- 

45^^  Ans.         gral  numbers, gives  45|^. 

20.  What  is  the  sum  of  3^   12|-,  and  25^?    A71S.  41|. 

21.  What  is  the  sum  of  -J,  15^,  42^,  and  50  ?/Vi  /  > 

22.  What  is  the  sum  of  30f,  1^,  leyV,  and  i|:? 

23.  Bought  3  pieces  of  cloth  containing  45|-,  384,  and 
35|-  yards  ;  how  many  yards  in  the  3  pieces  ? 

Ans.  119 1^2"  yards. 

24.  Three  men  bought  a  horse.  A  paid  31|  dollars,  B 
paid  43^  dollars,  and  C  paid  47|-  dollars  ;  what  was  the 
cost  of  the  horse  ?  Ans.  122|- dollars. 

25.  If  it  take  5-|-  yards  of  cloth  for  an  overcoat,  4^ 
yards  for  a  dress  coat,  2J  yards  for  a  pair  of  pantaloons, 
and  -|  of  a  yard  for  a  vest,  how  many  yards  of  cloth  will 
it  take  for  the  whole  suit  ?  Ans.  12|-  yards. 


86  FRACTIONS. 

SUBTRACTION. 

96.  The  process  of  subtracting  one  fraction  from  an- 
other is  based  upon  the  following  principles  : 

I.  One  number  can  be  subtracted  from  another  only 
when  the  two  numbers  have  the  same  unit  yalue.    Hence, 

II.  In  subtraction  of  fractions,  the  minuend  and  sub- 
trahend must  have  a  common  denominator. 

1.  From  Y^^  subtract  ^^-o. 

Analysis.  Since  the  fractions  have  a  common  denominator,  the 
difference  is  obtained,  by  subtracting  the  less  numerator  5,  from  the 
greater  9,  and  writing  the  result  over  the  common  denominator  13 ; 
we  thus  obtain  -^^,  the  required  difference. 

2.  From  |  subtract  f . 

3.  From  \\  subtract  y^- 

4.  Subtract  -|^  from  f^. 

5.  James  had  -J  of  a  bushel  of  walnuts,  and  ^CA  f  of 
them  ;  how  many  had  he  left  ? 

6.  Harvey  had  if  of  a  dollar,  and  gave  -ft  oi  ^  dollar 
to  a  beggar  ;  what  part  had  he  left  ?  \ 

7.  Subtract  -f  from  f . 

OPERATION.  Analysis,     ii*^  the  given  frac- 

2       3  —  14 .9_  —  _5_     Anro      tions  have  not  a  common  denom- 

inator,  first  reduce  them  to  the 
same  fractional  unit,  (1>4)  and  then  subtract  the  less  numerator  9, 
from  the  greater  14,  and  write  the  result  over  the  common  de- 
nominator 31.     We  thus  obtain  4j,  the  required  difference^ 

Rule.  I.  When  the  fractions  have  the  safne  denomina- 
tor, subtract  the  less  numerator  from,  the  greater,  and 
place  the  result  over  the  common  denominator. 

II.  When  they  have  not  a  common  denominator,  reduce 
them  to  a  common  deno7ninator  before  suUracting. 


Ans 

■  h 

Ans 

■  1- 

Ans. 

if 

Ans. 

H- 

Ans. 

7 
3  6- 

Ans. 

7 
2  0- 

Ans. 

3 
C6- 

Ans. 

1  9 

3T- 

Ans. 

^• 

Ans. 

xV 

s  u  b  t  k  a  c  t  i  o  n . 

Examples  for  Practice. 

8.  From  -J  take  f . 

9.  From  f  take  f. 

10.  From  |  take  |. 

11.  From  -^Q  take  |-.  • 

12.  Subtract  f  from  f. 

13.  Subtract  ^  from  f. 
■  14.  Subtract  |-  from  ^. 

15.  Subtract  y''^  from  {^, 

16.  Subtract  -|^  from  f. 

17.  Subtract  f^  from  -J. 

18.  From  9^  take  2|.  ^  - 

OPEKATiON.  Analysis.     First  reduce  the  fractional 

94-=:  9 Y^  parts,  ^  and  f,  to  a  common  denominator 

2^— -2-JL.  12.     Since  we  cannot  take  -^^  from  j^^-,  we 

add  1=^  to  y%,   which  makes  ^|,  and 

63^,  Ans.       y^a  from  |-|  leaves  j'^.      We  now  add  1  to 
the  2  in  the  subtrahend,  and  say,  3  from 
9  leaves  6.     We  thus  obtain  G^j,  the  difiference  required. 

Hence,  to  subtract  mixed  numbers,  we  may  redicce  the 
fractional  parts  to  a  common  denominator,  ayid  then  sub- 
tract the  fractional  and  integral  parts  separately. 

19.  From  24|  take  Yl\.  Ans.  7f 

20.  From  147|  take  49|-.  Ans.  9S^, 

21.  From  75^  take  40|.  A71S.  ^4:^. 

22.  From  63^^  take  22f.  Ans.  40|-. 

23.  Bought  flour  at  6^  dollars  a  barrel,  and  sold  it  at 
7f  dollars  a  barrel ;  what  was  the  gain  per  barrel  ? 

A71S.  ^  of  a  dollar. 

24.  From  a  cask  of  wine  containing  38f  gallons,  15| 
gallons  were  drawn  ;  how  many  gallons  remained  ? 

A?is.  22 1^  gallons. 


88  FEACTIOKS. 

MULTIPLICATION, 

Case  I. 
97.  To  multiply  a  fraction  by  an  integer. 

1.  If  1  pound  of  sugar  cost  ^  of  a  dollar,  what  will  3 
pounds  cost? 

Analysis.  If  1  pound  cost  |  of  a  dollar,  3  pounds,  whicli  are  3 
times  1  pound,  will  cost  3  times  |,  or  |-  of  a  dollar.  Therefore,  3 
pounds  of  sugar,  at  ^  of  a  dollar  a  pound,  will  cost  f  of  a  dollar. 

2.  If  1  liorse  eat  |-  of  a  ton  of  hay  in  1  month,  how 
much  will  4  horses  eat  in  the  same  time  ? 

3.  At  f  of  a  dollar  a  bushel,  what  will  be  the  cost  of  2 
bushels  of  pears  ?  of  3  bushels  ?  of  5  bushels  ? 

4.  How  many  are  3  times  f  ?  5  times  f  ?  4  times  ^  ? 
6  times  f  ?  9  times  yV  ^  ^  times  |  ? 

5.  If  one  yard  of  cloth  cost  -|  of  a  dollar,  what  will  3 
yards  cost  ? 


FIRST  OPERATION.  ANALYSIS.     In  the  first  Operation  we 

|.x3=:-^  =  2i-  multiply  the  fraction  by  3,  by  multi- 

„^^^.  ^  ^^^    ,.„  ^,^       plyii^S:  its   numerator  by  3,  obtaiiiins: 

SECOND  OPERATION.        ^  •'      »  -^        '  ^ 

g      o 5 f)j  i^=2^.     In  this  case  the  value  of  the 

6  2         ^  fractional  unit  remains  the  same,  but 

we  multiply  the  number  taken,  3  times.  In  the  second  operation  we 
multiply  the  fraction  by  3,  by  dividing  its  denominator  by  3,  obtain- 
leg  |=2|-.  In  this  case  the  value  of  the  fractional  unit  is  multi- 
plied 3  times,  but  the  nurnher  taken  is  the  same.     Hence, 

Multiplying  a  fraction  consists  in  multiplying  its  nu' 
merator,  or  dividing  its  denominator. 

Always  divide  the  denomiuator  when  it  is  esactly  divisible  by  the  multiplier. 


MULTIPLICATION. 


89 


Examples  for  Practice. 


6.  Multiply  I 

7.  Multiply  I 

8.  Multiply  ^ 


by  5. 
by  4. 
by  6. 


42  ■ 


9.  Multiply  if  by  9. 

10.  Multiply  ^4  by  3. 

11.  Multiply  If  by  14. 

12.  Multiply  41  by  5. 


A  ns. 
Ans, 
A?is. 
Ans. 
Ans. 
Ans.  10 


4. 


OPERATION. 


4i 
5 

If 
20 

21f 


Or, 


4i-v 


fraction,  and  tb  an  multiply  it. 


Analysis.  To  multiply  a  mixed 
number,  tt^  first  multiply  the 
fractional  part,  then  the  integer, 
and  then  add  the  two  products. 
Thus,  5  X  i=t=lf ;  and  5  X  4=20, 
which  added  to  If,  gives  21 1,  the 
required  result.  Or,  reduce  the 
mixed    number    to    an   imi^rope? 


Ans.  54. 


A71S.  68|. 


13.  Multiply  6|  by  8. 

14.  Multiply  9f  by  7. 

15.  If  a  man  earn  1-}  dollars  in  1  day,  liow  much  will 
lie  earn  in  10  days  ?  Ans.  IBf  dollars. 

16.  What  will  14  yards  of  cloth  cost,  at  f  of  a  dollar  a 
yard?  Ans.  10  dollars. 

17.  At  3^  dollars  a  cord,  what  will  be  the  cost  of  20 
cords  of  wood  ?  Ans.  65  dollars. 

18.  If  one  man  can  mow  l^V  acres  of  gra^s  in  a  day, 
how  many  acres  can  5  men  mow?  Ans.   9|-  acres. 

19.  What  will  9  dozen  eggs  cost,  at  14|-  cents  a  dozen  ? 

*  A71S.  130^  cents. 

20.  At  6f  dollars  a  barrel,  what  mil  30  barrels  of  flour 
cost  ?  Ans.  204  dollars. 


90 


FRACTIONS. 


Case  II. 

98.  To  multiply  an  integer  by  a  fraction. 

1.  At  9  dollars  a  barrel,  what  will  f  of  a  barrel  of  floui 


cost? 


Since  1  barrel  of  flour  cost  9  dollars,  f  of  a  barrel 


Anjllysis 

will  cost  2  times  |-  of  9  dollars.     |^  of  9  dollars  is  3  dollars^  and  |  of 
9  dollars  is  2  times  3  dollars,  or  6  dollars 
will  cost  6  dollars. 


Therefore  f  of  a  barrel 


2.  If  a  yard  of  clotli  be  worth  8  dollars,  what  is  f  of  a 
yard  worth  ? 

3.  If  an  acre  of  land  produce  25  bushels  of  wheat,  how- 
much  will  \  of  an  acre  produce  ?  f  of  an  acre  ?  f  of  an 
acre  ? 

4.  If  a  man  earn  20  dollars  in  a  month,  what  can  he 
earn  in  -J-  of  a  month  ?  in  |  ?  in  y^  ?  in  f  ? 

6.  If  a  ton  of  hay  cost  12  dollars,  what  will  J  of  a  ton 
cost?  I  of  a  ton  ?  f  of  a  ton  ?  |-  of  a  ton  ? 

6.  At  60  dollars  an  acre,  what  will  f  of  an  acre  of  land 
cost? 


FIKST  OPERATION. 
5  )  60   price  of  1  acre. 

1 2   cost  of  J  of  an  acre. 

4 

48   cost  of  f  of  aB  acre. 

SECOND  OPERATION, 

60  price  of  1  acre. 

4 

5  )  240  cost  of  4  acres. 

48  cost  of  f  of  an  acre. 


Analysis.  4  fifths  of  an  acre 
will  cost  4  times  as  much  as  1  fifth 
of  an  acre,  or  4  times  i,  of  60  dol- 
lars. ^  of  60  dollars  is  12  dollars, 
and  I  is  4  times  12,  or  48  dollars, 
the  cost  of  f  of  an  acre.  In  the 
second  operation,  we  multiply  the 
price  of  1  acre  by  4,  and  obtain 
240  dollars,  the  cost  of  4  acres  ; 
but  as  f  of  1  acre  is  the  same  as 
^  of  4  acres,  we  divide  240  dol- 
lars,  the  cost  of  4  acres,  by  5,  and 
obtain  48  dollars,  the  cost  of  f  of 
an  acre. 


MULTIPLICATIOliJ".  91 

Rule.  Multiplying  an  integer  ly  a  fraction,  consists 
in  multiplying  ly  the  numerator,  and  dividing  the  product 
hy  the  denominator. 

7.  Multiply  45  by }.  ^  Ans.  33J. 

8.  Multiply  68  by  f  '  Ans.  54|. 

9.  Multiply  105  by  yV  Ans.     49. 

10.  Multiply  480  by  |.  Ans.  300. 

11.  At  16  dollars  a  ton,  what  will  be  the  cost  of  f  of  a 
ton  of  hay  ?  Ans.  VZ  dollars. 

12.  If  a  village  lot  is  worth  340  dollars,  what  is  J  of  it 
"vvorth?  ^«5.  255  dollars. 

13.  If  a  hogshead  of  sugar  is  worth  75  dollars,  what  is 
\\  of  it  worth  ?  A71S.  68J  dollars. 

14.  If  an  acre  of  land  produce  114  bushels  of  oats,  how 
many  bushels  will  -^^g  of  an  acre  produce  ? 

Ans.  64i  bushels. 

15.  If  a  man  trav'^1 47  miles  in  a  day,  how  far  does  he 
travel  in  -|  of  a  day  ?  Ans.  29f  miles. 

Case  III. 

99.  To  multiply  a  fraction  by  a  fraction. 

1.  If  a  bushel  of  apples  is  worth  J  of  a  dollar,  what  is 

J  of  a  bushel  worth  ? 

Analysis.  Since  1  bushel  is  worth  \  of  a  dollar,  |^  of  a  hushel 
is  worth  4  times  ^  of  a  dollar ;  \  equals  f ,  and  4  of  f  is  ^.  There- 
fore I  of  a  bushel  is  worth  |  of  a  dollar. 

2.  If  a  yard  of  cloth  cost  -J  a  dollar,  what  will  ^  of  a 
yard  cost  ? 

3.  When  oats  are  worth  J-  of  a  dollar  a  bushel,  what  is 
J  of  a  bushel  worth  ? 

4.  If  a  man  own  ^  of  a  vessel,  and  he  sells  ^  of  his 
share,  what  part  of  the  vessel  does  he  sell  ? 


r 


93  FRACTIONS. 

5.  At  f  of  a  dollar  a  bushel,  what  will  f  of  a  bushel  of 
corn  cost  ? 

OPERATioisr.  Analysis.     Since  1  bushel  cost 

■|xf  =  3^3=^,  A71S.  f  of  a  dollar,  f  of  a  bushel  will 

cost  f  times  |  of  a  dollar.  By 
multiplying  the  numerators  2  and  3  together,  we  obtain  the  numer- 
ator 6  of  the  product  ;  and  by  multiplying  the  denominators  3  and 
4  together,  we  obtain  the  denominator  12  of  the  product,  and  thus 
we  have  I'^g  =  I  for  the  required  product.   Hence  we  have  the  following 

Rule.  Multiply  togefhei'  the  mimerators  for  a  new  mi- 
merator,  and  the  denominators  for  a  neiv  denominator, 
and  reduce  the  result  to  its  loivest  terms. 


_3_ 

1  0* 

_9_' 

2  8* 


Examples  eor  Peactice. 

6.  Multiply  4  by  |.  Ans. 

7.  Multiply  f  by  f  Ans. 

8.  Multiply  f  by  -f.  Ans.  ff 

9.  Multiply  I  by  f.  Ans.  ^^. 

10.  Multiply  yV  ^y  f  -^^^^-  A- 

11.  What  is  the  product  of  |,  J-  and  J?         Ans.  jV 

12.  What  is  the  product  of  f,  -|-  and  f  ?         Ans.  -f-^. 

13.  What  is  the  product  of  f,  f  and  -^  ?      Ans.     ^. 

14.  What  is  the  product  of  ff  and  If  ?         Ans.     ^. 

15.  What  is  the  product  of  f ,  1^,  5  and  |  ? 

OPERATION.  When    integers    or 

|.  X  1|-  X  5  X  |-=  mixed  nu^nhers    occur 

fx   f  xfx|=''"r"— ^^  ^^^^-         among  the  given  fac- 
tors, they  may  be  re- 
duced to  improper  fractions  before  multiplying ;  and  an 
integer  may  be  reduced  to  the  form  of  a  fraction  by  writ- 
ing 1  for  its  denominator  ;  thus,  5=|-. 

16.  What  is  the  product  of  f,  f  and  2|  ?      Ans.  ^f. 

17.  What  is  the  product  of  3,  yV  '^^^^  I?      ^^5-  ^i- 


MULTIPLICATION.  93 

18.  What  is  the  product  of  f ,  -fj  and  f|  ?    Ans.  ■^. 

19.  Find  the  Takie  of  f  of  -f  multipHed  by  |  of  -f^. 

OPERATION. 

f  X I  X  -i  X  ^=^,  Ans. 

1.  Fraction?  -with  the  word  of  between  them  are  sometimes  called  compound 
fractious.  The  word  of  is  pimply  an  equivalent  for  the  sign  x  of  muliiplica- 
tioa.  and  signifies  that  the  numbers  between  which  it  is  placed  are  to  be  multi- 
plied together, 

2.  When  th3  same  factors  occur  in  both  numerator  and  denominator  of  frac- 
tions to  be  multiplied  together,  they  may  be  omitted  and  the  remaining  factors 
only  used  ;  thus,  5  and  3  being  found  in  both  the  numerators  and  denominators 
of  the  above  esample  may  be  omitted  in  multiplying. 

20.  Multiply  f  of  f  by  |  of  -J-.  Ans.  ^V 

21.  Multiply  f  of  3  by  -J  of  2f  Ans.  Sf 

22.  What  is  the  product  of  -{■^,  J  of  f  and  1^  ? 

^ns.  -Q-. 

23.  What  is  the  product  of  f  of  y\  by  5-J-?      Ans.  3. 

24.  What  is  the  value  of  |  times  -J-  of  f  of  10  ? 

yLns.  -g-. 

25.  What  is  the  value  of  ^  of  f  times  J  of  3f  ? 

Ans.  f. 

26.  At  f  of  a  dollar  a  bushel,  what  will  f  of  a  bushel  of 
corn  cost  ?  Ans.  ^  of  a  dollar. 

27.  When  j^eaches  are  worth  -^-^  of  a  dollar  a  bushel, 
what  is  f  of  a  bushel  worth  ?  Ans.  ^  dollar. 

28.  Jane  having  f  of  a  yiird  of  silk  gave  f  of  it  to  her 
sister  ;  what  part  of  a  yard  did  she  give  her  sister  ? 

Ans.  f  of  a  jSLvd. 

29.  When  pears  are  worth  -|  of  a  dollar  a  basket,  what 
is  -f-  of  I  of  a  basket  worth  ?  A?is.  f  of  a  dollar. 

30.  A  man  owning  -Jf  of  a  ship,  sold  f  of  his  share  ; 
what  loart  of  the  whole  ship  did  he  sell?  Ans.  ^. 

31.  A  grocer  having  if  of  a  hogshead  of  molasses  sold 
•^^g-  of  it  ?  what  part  of  a  hogshead  remained  ? 

32.  At  f  of  a  dollar  a  yard,  what  will  be  the  cost  of  f  of 
3  yards  of  cloth  ?     .  Aiis.  1}  dollars. 


94  FEACTIONS. 

DIVISION. 

Case  I. 

100.  To  divide  a  fraction  by  an  integer. 

1.  If  3  pounds  of  raisins  cost  |-  of  a  dollar,  what  will 
1  pound  cost? 

Analysis.  If  3  pounds  cost  |^  of  a  dollar,  1  pound,  whicli  is 
^  of  3  pounds,  will  cost  ^  of  ^,  or  ^  of  a  dollar.  Tlierefore,  1  pound 
will  cost  %  of  a  dollar. 

2.  If  4  pounds  of  coffee  cost  -J  of  a  dollar,  what  will 
1  pound  cost  ? 

3.  If  5  marbles  cost  -|  of  a  dollar,  what  will  1  marble 
cost  ? 

4.  If  -J  of  a  barrel  of  flour  is  equally  divided  among 
6  persons,  what  part  of  a  barrel  will  each  hh;ve  ? 

5.  If  4"  of  a  box  of  tea  is  equally  distribited  among 
8  persons,  what  part  of  a  box  will  each  have  ? 

6.  Paid  f  of  a  dollar  for  4  pounds  of  butter ;  what  was 
the  cost  per  pound  ? 

FIRST  OPERATION.  ANALYSIS.      In    the    first    operation 

I -4-4=1,  Aus.  'W'e  divide  the  fraction  by  4,  by  divid- 

ing its   numerator  by   4,   obtaining  f. 
SECOND  OPERATION.       In  this  case  the  value  of  the  fractional 
-|-^4=:-j^g-rr:|,  Aus.     ^^nit   is   unchanged,    but   we    diminish 

the  number  taken,  4  times.  In  the 
second  operation  we  divide  the  fraction  by  4,  by  multiplying 
the  denominator  by  4,  obtaining  j\=f.  In  this  case  the  value 
of  the  fractional  unit  is  diminished  4  times,  but  the  nuniber  taken 
is  the  same.    Hence, 

Dividing  a  fraction  consists  in  dividing  its  numerator, 
or  multiplying  its  denominator. 

We  divide  Ihe  numerator  when  it  is  exactly  divisible  by  the  divisor;  other 
wise  we  multiply  the  denominator. 


division".  95 

Examples  for  Peactice. 

7.  DmdefbyS. 

8.  Divide  f  by  4. 

9.  Divide -}i  by  5. 

10.  Divide  ^  by  5. 

11.  Divide  i  by  9. 

12.  Di^dde  If  by  21. 

13.  Divide  f  of  I  by  12. 

14.  Divide  f  of  f  by  6. 

15.  Di\dde  4|  by  7. 

OPERATION. 

4^;^2gL  Reduce  the  mixed  number  to  an 

2 1  _i_'j'— -  3    jifis.        improper  fraction  and  then  divide  as 


Ans. 

f- 

Ans. 

i- 

Ans. 

2 

IS- 

Ans. 

3 
16" 

Ans. 

1 
"S-3"- 

Ans. 

A. 

Ans. 

A- 

Ans. 

A- 

before. 


1  0 
20' 


3. 

8* 

2  9 
40* 


16.  Divide  3f  by  4.  iS^zs. 

17.  Divide  6|  by  9.  -^?Z5.  H. 

18.  Divide -|- of  2J  by  3.  ^?2.«?. 

19.  Divide  ^^  by  12.  ^«s. 

20.  Divide  13|  by  10.  Ans.  1|. 

21.  Divide  |  of  8'by  20.  Ans.   J. 

22.  If  6  persons  agree  to  share  equally  f  of  a  bushel  of 
grapes,  what  part  of  a  bushel  will  each  have  ?    Ans.  -J. 

23.  If  5  yards  of  sheeting  cost  ^  of  a  dollar,  what  will 
1  yard  cost  ?  Ans.  -f^  of  Oi^  dollar. 

24.  If  8  bushels  of  apples  cost  5|-  dollars,  what  will  1 
bushel  cost  ?  ^7^5.  |  of  a  dollar. 

^  25.  If  -J-  of  10  pounds  of  butter  cost  IJ  dollars,  what 
will  1  pound  cost  ?  Ans.  |-  of  a  dollar. 

26.  A  man  distributed  ^§-  of  a  dollar  equally  among  6 
beggars  ;  what  part  of  a  dollar  did  he  give  to  each  ? 

27.  If  f  of  9  cords  of  wood  cost  12|  dollars,  what  will 
1  cord  cost  ? 


96  FRACTIONS. 

Case  II. 

101.  To  divide  an  integer  bj  a  fraction. 

1.  At  -|^  of  a  dollar  a  yard,  how  many  yards  of  ribbon 
can  be  bought  for  2  dollars  ? 

Analysis.  As  many  yards  as  -|-  of  a  dollar,  the  price  of  1  yard, 
is  contained  times  in  2  dollars.  Since  in  1  dollar  there  are  3  thirds 
of  a  dollar,  in  two  dollars,  there  are  2  times  3  thirds,  or  6  thirds; 
and  1  third  is  contained  in  6  thirds,  6  times.  Therefore  6  yards  of 
ribbon  can  be  bought  for  2  dollars. 

2.  When  potatoes  are  f  of  a  dollar  a  bnshel,  how  many 
bushels  can  be  bought  for  2  dollars  ?  For  4  dollars  ?  For 
6  dollars  ? 

3.  If  a  man  spend  -^  of  a  dollar  a  day  for  cigars,  how  long 
will  it  take  him  to  spend  3  dollars  ?  5  dollars  ?  6  dollars  ? 

4.  At  |-  of  a  dollar  a  bushel,  how  many  bushels  of  corn 
can  be  bought  for  16  dollars  ; 

FIKST  OPERATION.  ANALYSIS.     As  many  bushels  as  f  of  a 

16  dollar,  the  price  of  1  bushel,  is  contained 

5  times  in  16  dollars.     But  we  cannot  divide 

.nTT  integers  by  fifths,  because  they  are  not 

'' —  of  the  same  denomination.     Reducing  16 

20  bushels.  dollars  to  fifths  by  multiplying  by  5,  we 

SECOND  OPERATION.  have  80  fifths,  and  4  fifths  is  contained 

4)16                 "  in  80  fifths,  20  times,  the  required  num- 

~T             .  ber  of  bushels.    In  the  second  operation, 

K  we  divide  the  integer  by  the  numerator 

of  the   fraction,   and   multiply  the   quo- 

20  bushels.  tient  by  the  denominator,  which  produces 
the  same  result  as  in  the  first  operation.    Hence 

Dividing  hy  a  fraction  consists  i^i  miiltiphjing  ly  the 
denominator,  and  dividing  the  iiroduct  hy  the  tiumeraior 
of  the  divisor. 


Ans. 

27. 

Ans. 

49. 

Ans. 

l^. 

Ans. 

m- 

Ans, 

220. 

Ans. 

136. 

Ans. 

627^ 

division-.  97 

Examples  for  Practice. 

5.  Divide    18  by    f. 

6.  Divide    14  by    f . 

7.  Di^dde    11  by    f.* 

8.  Divide    75  by  yV 

9.  Divide  120  by  ^. 

10.  Di^dde    96  by  if. 

11.  Divide  226  by  ^j. 

12.  Divide    28  by  4|. 

OPERATION.  Analysis.     Reduce  tlie  mixed  num- 

23v^3— -g4  ber  to  an  improper  fraction,  and  then 

84-i-14rzr6  Ans.  divide  the  integer  in  the  same  manner 

as  by  a  proper  fraction. 

13.  Divide    16  by  2^.  Ans.  ^. 

14.  Divide    42  by  3J.  Ans.  12. 

15.  Di^dde  112  by  6|.  A^is.  17^. 

16.  Divide  180  by  7^.  Ans.  25^. 

17.  Di\dde425  by    f.  Ans.  595, 

18.  Divide  318  by  A-  ^^^-  l^^^- 

19.  When  potatoes  are  -J  of  a  dollar  a  bnsbel,  how  many 
bushels  can  be  bought  for  10  dollars?     Ans.  12f  bush. 

20.  Divide  9  bushels  of  corn  among  some  persons,  gi^*ing 
them  y\  of  a  bushel  each  ;  how  many  persons  will  receive 
a  share  ?  Ans.  48. 

21.  At  2f  dollars  a  cord,  how  many  cords  of  wood  can 
be  bought  for  27  dollars  ?  Ans.  9y^j  cords. 

22.  If  a  horse  eat  f  of  a  bushel  of  oats  in  a  day,  in 
how  many  days  will  he  eat  20  bushels  ?     Aiis.  36  days. 

23.  If  a  man  walk  ^i-f^  miles  an  hour,  how  many  hours 
will  he  require  to  walk  48  miles?         Ans.  16^  hours. 

24.  At  -jig-  of  a  dollar  a  pound,  how  many  pounds  of  rice 
can  be  bought  for  3  dollars  ?  A'ns.  48  pounds. 

7 


98  FEACTIONS. 

Case  III. 
103.  To  divide  a  fraction  by  a  fraction. 

1.  At  f  of  a  dollar  a  pound,  how  many  pounds  of  tea 
can  be  bought  for  f  of  a  dollar  ? 

Analysis.  As  many  pounds  as  f  of  a  dollar,  the  price  of  1 
pound,  is  contained  times  in  |  of  a  dollar  ;  2  fifths  are  contained  in 
A  fifths,  2  times.  Therefore  2  pounds  can  be  bought  for  |  of  a 
dollar. 

Hence  we  see,  that  when  fractions  have  a  common  denominator, 
division  may  be  performed  by  dividing  the  numerator  of  the  divi- 
dend by  the  numerator  of  the  divisor. 

2.  How  many  pine-apples  at  -f^  of  a  dollar  each,  can  be 
bought  for  -f^  of  a  dollar  ?  for  y%  ?  f or  -fl  ? 

3.  If  a  horse  eat  f  of  a  bushel  of  oats  in  1  day,  in  how 
many  days  will  he  eat  ^  of  a  bushel  ?  f  ?  -^  ?  ^  ? 

4.  At  -J-  of  a  dollar  a  bushel,  how  many  bushels  of  ap- 
ples can  be  bought  for  f  of  a  dollar  ?  for  |-  ?  f or  f  ? 

5.  At  f  of  a  dollar  a  pound,  how  many  pounds  of  tea 
can  be  bought  for  f  of  a  dollar  ? 

FIRST  OPERATION.  ANALYSIS.  Asmanypounds 

■|=zA-  ;  |-=:44.  as  |-  of  a  dollar,  the  price  of  1 

15.    8  IT      J400  pound,  is  contained  times  in 

f  of  a  dollar,     t  equal  Jn,  f 
SECOND  OPERATION.  *        ,    ,  ^  ,   ^  .    7 

^^g.—  3v5.— i5_li    y|.,Q      equal  -If,  and  8  twentieths  are 
4  .  r— T^2—  ^— -^-J?  ^'^*-     contained  in  15  ticentieths  1| 

times.  Or,  as  in  the  second  operation,  we  have  multiplied  the  divi- 
dend f  by  the  denominator  5,  of  the  divisor,  and  divided  the  result 
by  the  numerator  2,  of  the  divisor.  Hence,  by  inverting  the  tenns 
of  the  divisor  the  two  fractions  will  stand  in  such  relation  to  each 
other,  that  we  can  multiply  together  the  two  upper  numbers  for  the 
numerator  of  the  quotient,  and  the  two  lower  numbers  for  the  de- 
nominator, as  shown  in  the  second  operation. 


Ans. 

3. 

Ans. 

2. 

Ans. 

H- 

Divisioif.  99 

EuLE.  I.  Reduce  integers  and  mixed  7iumbers  to  inv 
proper  fractions. 

11.  Invert  the  terms  of  the  divisor,  and  proceed  as  in 
multiplicatio7i. 

Examples  foe  Peactice. 

6.  Diyide  i^  by  ■^.  « 

7.  Divide  ^  by  J. 

8.  Divide  |  by  f. 

9.  Divide  |  by  f .  Ans.  2^. 

10.  How  many  times  is  ^  contained  in  |^  ?  A7is.  2f . 

11.  How  many  times  is  -f-  contained  in  f  ?     Ans.  -^. 

12.  How  many  times  is  \  contained  in  ^  ?   Ans.  1-|. 

13.  Divide  i  of  J  by  f.  Ans.  -f. 

14.  Divide  i  of  -|  by  -^.  Ans.  If. 

15.  DiA-ide  \\  by  \  off.  Ans.  7^. 

16.  Divide  J  of  -J  by  f  of  -J.  Ans.  l^t- 

17.  At  4  of  a  dollar  a  pound,  how  many  pounds  of  sugar 
can  be  bought  for  |-  of  a  dollar?  Ans.  5f  pounds. 

18.  At  -^  of  a  dollar  a  pint,  how  much  wine  can  be 
bought  for  |-  of  a  dollar  ?  Ans.  -f  of  a  pint. 

19.  At  f  of  -J  of  a  dollar  a  yard,  how  many  yards  of 
ribbon  can  be  bought  for  ^-^  of  a  dollar  ?  Ans.  2f  yards. 

20.  At  |-  of  a  dollar  a  yard,  how  many  yards  of  silk  can 
be  bought  for  f  of  a  dollar?  Ans.  %\  yards. 

21.  A  man  owning  -|  of  a  copper  mine,  divided  his  share 
equally  among  his  sons,  giving  them  -f^  each  ;  how  many 
sons  had  he  ?  Ans.  2. 

22.  If  -f-  of  a  bushel  of  pears  cost  f  of  a  dollar,  what 
will  1  bushel  cost  ?  Ans.  ^  of  a  dollar. 

23.  How  much  corn  at  f  of  a  dollar  a  bushel,  can  be 
bought  for  f  of  a  dollar.  Aris.  f  of  a  bushel. 


100  '  fractions. 

Promiscuous  Examples.  , 

1.  In  25-fj  i^oimds  how  many  16ths  of  a  pound  ?/ 

2.  Reduce  ^f^  to  a  mixed  number.  Ans.  11-||-. 

3.  Eeduce  l^ff  to  its  lowest  terms.  Ans.  f . 

4.  In  ■^\—  of  a  day  how  many  days  ? 

5.  Change  42  ponnds  to  sevenths  of  a  pound. 

6.  Eeduce  21|-  to  an  improper  fraction.      Ans.  J-|^. 

7.  Reduce  126f  to  thirds.  Ans.  ^^. 

8.  Reduce  fff  to  its  lowest  terms.  Ans.  ^. 

9.  Reduce  |-  and  f  to  a  common  denominator. 

10.  Reduce  36  to  a  fraction  whose  denominator  is  12. 

11.  What  is  the  sum  of  |,  |  and  |  ?  Ans.  1^. 

12.  Add  together  -^^  -|-  and  3-|-.  Ans.  4f. 

13.  What  is  the  difference  between  -J  and  f  ? 

14.  Reduce  -^,  -J  and  f  to  a  common  denominator. 

15.  Sold  9|  cords  of  wood  to  one  man,  and'12YVto 
another  ;  how  much  did  I  sell  to  both  ? 

16.  Paid  STf'o"  dollars  for  a  horse,  and  62-|-  dollars  for  a 
wagon  ;  how  much  more  was  paid  for  the  horse  than  for 
the  wagon  ?  Ans.  25|-  dollars. 

17.  A  farmer  having  234^  acres  of  land,  sells  at  one 
time  42f  acres,  at  another  time  61f,  and  at  another  70| 
acres  ;  how  many  acres  has  he  left  ?    Ans.  60y\-  acres. 

18.  A  speculator  bought  120  bushels  of  wheat,  for  lo6f 
dollars,  and  sold  it  for  197f  dollars  ;  what  did  he  gain? 

19.  Bought  12  pounds  of  coffee  at  ^  of  a  dollar  a  pound, 
and  9  pounds  of  tea  at  f  of  a  dollar  a  pound ;  what  was 
the  cost  of  the  whole  ?  Ans.  8^  dollars. 

20.  Bought  10  bushels  of  wheat,  at  l-J  dollars  a  bushel, 
and  14  bushels  of  corn,  at  -f-  of  a  dollar  a  bushel ;  which 
cost  the  more,  and  how  much  ? 

Ans.  The  wheat,  3|^  dollars. 


PEOMISCUOUS     EXAMPLES.  101 

21.  Paid  12  dollars  for  some  cloth,  at  the  rate  of  f  of  a 
dollar  a  yard  ;  how  many  yards  were  purchased  ?     ^o*^  *  - 

%%.  If  8  oranges  cost  f  of  \\  dollars,  what  will  1  orange 
cost?  Ans.  -^  oi  2^  ^oViSiY, 

23.  A  man  bought  |-  of  a  farm  and  sold  f  of  his  share  ; 
what  part  of  the  w^hole  farm  did  he  sell  ?  what  part  had 
he  left?  Ans.  Sold  ^. 

24.  If  a  barrel  of  sugar  is  worth  22  dollars,  what  is  -^ 
of  it  worth  ?  A^ns.  15|  dollars. 

25.  How  many  hours  will  it  take  a  man  to  travel  136 
miles,  if  he  travel  3f  miles  an  hour  ?    Ans.  41^  hours. 

26.  How  many  barrels  of  apples  can  be  bought  for  18 
dollars,  at  1^  dollars  a  barrel?  Ans.  ISy^g-  barrels. 

27.  If  the  smaller  of  two  fractions  be  -^,  and  the  differ- 
ence f ,  what  is  the  greater  ?  Ans.  |-|. 

28.  If  the  sum  of  two  fractions  is  \\,  and  one  of  them 
is  ^,  v\^hat  is  the  other?  ,,  Ans.  |^. 

-  29.  If  the  dividend  be  ff,  and  the  quotient  f,  what  is 
the  di\isor  ?  Ans,  \\. 

30.  If  the  divisor  be  -j^,  and  the  quotient  3  J,  what  is  the 
dividend?  Ans,  2y*g-. 

31.  How  many  bushels  of  oats  worth  f  of  a  dollar  a 
bushel,  will  pay  for  f  of  a  barrel  of  flour  worth  9  dollars 
a  barrel  ?  Ans.  15  bushels. 

32.  At  -|  of  a  dollar  a  rod,  what  will  it  cost  to  dig  \  of 
f  of  h\  rods  of  ditch  ?  Ans.  -^  dollars. 

33.  If  a  man  has  24J  bushels  of  clover  seed,  and  he  sells 
I"  of  it,  how  much  has  he  left  ?  Ans.  6^  bushels. 

34.  A  man  had  6  lots  of  land,  each  containing  37 J 
acres  ;  how  many  acres  did  they  all  contain  ? 

35.  If  f  of  a  ton  of  hay  can  be  bought  for  15  dollars, 
what  part  of  a  ton  can  be  bought  for  1  dollar  ? 


1052  DECIMALS. 

DECIMAL  FEAOTIONS. 

NoTATio:^  AND  Numeration. 

103.  Decimal  Fractions  are  fractions  wliicli  liaye  for 

their  denominator  10,  100, 1000,  or  1  with  any  number  of 

ciphers  annexed. 

Decimal  fractions  are  commonly  called  decimals. 
Since  -^  =  yVV  ih  —  two:»  ©tc,  the  denominators  of 

decimal  fractions  increase  and  decrease  in  a  tenfold  ratio, 

the  same  as  simple  numbers. 

104o  In  the  formation  of  Decimals  a  unit  is  divided 
into  10  equal  parts,  called  tenths  ;  each  of  these  tenths  is 
divided  into  10  other  equal  parts  called  hundredths;  each 
of  these  hundredths  into  10  other  equal  parts,  called  thou- 
sandths, and  so  on.  Since  the  denomini^tors  of  decimal 
fractions  increase  and  decrease  by  the  scale  of  10,  the 
same  as  simple  numbers,  in  writing  decimals  the  denom- 
inators may  be  omitted. 

105.  The  Decimal  sign  (.)  is  always  placed  before 
decimal  figures  to  distinguish  them  from  integers.  It  is 
commonly  called  the  decimal  point.     Thus, 

-j^    is  expressed  .6 

_i_4_    'i        <^         .54 

10  0  '^^ 

27  9       i<  a  970 

1  0  0  0  "^*  "^ 

.5      is  5  tenths,  which  =  ^  of  5  units  ; 

.05     is  5  hundredths,  *'      =  -jig-  of  5  tenths  ; 

.005  is  5  thousandths,  "      =  -Jg-  of  5  hundredths. 

And  universally,  the  value  of  a  figure  in  any  decimal 
place  is  -^  the  value  of  the  same  figure  in  the  next  left 
hand  place. 


NOTATIOlf     AN"D     KU3IERATI0N.  103 

106.  The  relation  of   decimals  and  integers  to  each 
other  is  clearly  shown  by  the  following 

Decimal  Numeeatioh  Table. 


• 

m 

dths 

»  S3 

^i 

^  ^ 

■^     QQ  , 

O      r- 

^-    .^-sg- 

Millions. 

Hand,  of  th 

Tens  of  thoi 

Thousands. 

Hundreds. 

Tens. 

Units. 

Decimal  8ig 

Tenths. 

Hundredths 

Thousandth 

Ten-thousan 

Hundred-th< 

Milhonths. 

573275  4.573256 

By 

exaniiniri 

g  this  table 

we  see  that 

Tenths 

are 

expressed  by  one  figure. 

Hundredths         '' 

'^          ^^  two  figures. 

Thousandths       ''  ''  ''  three     '' 

107.  Since  the  denominator  of  tenths  is  10,  of  hun- 
dredths 100,  of  thousandths  1000,  and  so  on,  a  decimal  may 
be  expressed  by  writing  the  numerator  only  ;  but  in  this 
case  the  numerator  or  decimal  must  always  contain  as  many 
decimal  places  as  are  equal  to  the  number  of  ciphers  in  the 
denominator ;  and  the  denominator  of  a  decimal  will 
always  be  the  unit  1,  with  as  many  ciphers  annexed  as  are 
equal  to  the  number  of  figures  in  the  decimal  or  numerator. 

The  decimal  point  jnust  never  be  omitted. 

Examples  tor  Peactice. 

1.  Express  in  figures  seyen-tenths.  Ans.  .7. 

2.  Write  tw^enty-five  hundredths.  Ans.  .25. 

3.  Write  nine  hundredths.  Ans.  .09. 
4  AVrite  one  hundred  twenty-fiye  thousandths. 

5.  Write  eighteen  thousandths. 


104  DECIMALS.      . 

6.  Write  fifty-eight  "hundredths. 

7.  Write  two  hundred  thirty-six  thousandths. 

8.  Write  one  thousand  three  hundred  twenty  ten- 
thousandths.  A71S.  .1320. 

9.  Write  seven  hundred  thirty-two  ten-thousandths. 

Eead  the  following  decimals  : 

.06  .143  .000  .479 

.34  .037  .3240  .00341 

.80  .472  .1026  .102367 

108.  A  mixed  number  is  a  number  consisting  of  inte- 
gers and  decimals  ;  thuS;,  71.406  consists  of  the  integral 
part,  71,  and  the  decimal  part,  .406  ;  it  is  read  the  same  as 
71^^^^,  71  and  406  thousandths. 

Examples  eor  Peactice. 

1.  Write  twenty-four,  and  four  tenths.      Ans.  24.4. 

2.  Write  thirty-two,  and  five  hundredths. 

3.  Write  seventy-six,  and  forty-six  thousandths. 

4.  Write  one  hundred  twelve,  and  one  hundred  ninety 
thousandths.  Ans.  112.190. 

5.  Write  sixty-three,  and  forty-four  ten-thousandths. 

6.  Write  seventy-five,  and  one  hundred  forty  ten-thou- 
sandths. 

7.  Write  five,  and  5  hundred-thousandths. 

8.  Write  sixteen,  and  21  ten-thousandths. 

9.  Write  eight,  and  234  hundred-thousandths. 

10.  AYrite  forty,  and  75  hundred-thousandths. 

A}is.  40.00075. 

11.  Eead  the  following  numbers : 


42.08 

50.002 

640.00010 

81.110 

161.0301 

7.4230 

120.0342 

14.42000 

3.01206 

NOTATION     AND     NUMERATION.  105 

109.  From  the  foregoing  explanations  and  illustra- 
tions we  deriye  the  following  important 

Principles  of  Decimal  !N"otation  and  Numeration. 

1.  The  value  of  any  decimal  figure  depends  upon  its 
place  from  the  decimal  point ;  thus  .3  is  ten  times  .03. 

2.  Prefixing  a  cipher  to  a  decimal  decreases  its  value  the 
same  as  dividing  it  by  ten  ;  thus  .03  is  yV  the  value  of  .3. 

2.  Annexing  a  cipher  to  a  decimal  does  not  alter  its 
value,  since  it  does  not  change  the  place  of  the  significant 
figures  of  the  decimal ;  thus,  -^,  or  .6,  is  the  same  as 
.^6  0     or  ()0 

10  0  ' 

4.  Decimals  increase  from  right  to  left,  and  decrease 
from  left  to  right,  in  a  tenfold  ratio  ;  and  therefore  they 
may  he  added,  subtracted,  multiplied,  and  divided  the 
same  as  whole  numbers. 

5.  The  denominator  of  a  decimal,  though  never  ex- 
pressed, is  always  the  unit  1,  with  as  many  ciphers 
annexed  as  there  are  figures  in  the  decimal. 

6.  To  read  decimals  requires  two  numerations ;  first, 
from  units,  to  find  the  name  of  the  denominator,  and 
second,  towards  units,  to  find  the  value  of  the  numerator. 

110.  Having  analyzed  all  the  principles  upon  which 
the  writing  and  reading  of  decimals  depend,  we  will 
now  present  these  principles  in  the  form  of  rules. 

EuLE  FOR  Decimal  I^otation. 

I.  Write  the  decimal  tJie  same  as  a  icliole  numler,  'plac- 
ing ciphers  where  necessary  to  give  each  significant  figure 
its  true  local  value. 

II.  Place  the  decimal  point  before  the  first  figure. 


106  DECIMALS. 

EuLE  FOR  Decimal  IN'umeeation'. 

I.  Numerate  from  the  decimal  point,  to  determine  the 
denominator. 

II.  Numerate  towards  the  decimal  point,  to  determine 
the  numerator. 

III.  Read  the  decimal  as  a  whole  numher,  giving  it  the 
name  of  its  lowest  decimal  unit,  or  right  hand  figure. 

Examples  foe  Peactice. 

1.  Write  325  ten-tlioiisandtlis.  Ans.  .0325. 

2.  Write  four  hundred  ten-tliousandths. 

3.  Write  117  ten-thousandths. 

4.  Write  ten  ten-thousandths.  Ans.  .0010. 

5.  Write  250  millionths.  Ans.  .000250. 

6.  Write  twelve  hundred  ten-thousandths. 

7.  Write  9  hundred-thousandths.  Ans.  .00009.    ♦ 

8.  Eead  the  following  decimals  : 

.1236  .00061  .32760 

.0080  .720000  .040721 

7.  Write  four  hundred,  and  nine  tenths. 

Ans.  400.9. 

10.  Write  twenty-seven,  and  fifty-six  hundredths. 

11.  Write  eighty-five,  and  one  hundred  fifty  thousandths. 

12.  Write  one  thousand,  and  twelve  millionths. 

13.  Write  three  hundred  sixty-five,  and  one  thousand 
eight  hundred  seven  hundred-thousandths. 

A?is.  365.01807. 

14.  Write  nine  hundred  ninety,  and  three  thousand  two 
hundred  fourteen  millionths.  Ans.  990.003214. 

15.  Read  the  follomng  numbers  : 

71.03  11.0003  34.800000 

126.326  240.01376  9.1263476 


EEDUCTIOlf.  107 

REDUCTION. 

Case  I. 
111.  To  reduce  decimals  to  a  common  denominator. 

1.  Reduce  .3,  .09,  .0426,  .214  to  a  common  denominator. 

OPERATION.        Analysis.     A  common  denominator  must  con- 

.3000  tain  as  mauy  decimal  places  as  is  equal   to  the 

.0900  greatest  number  of  decimal  figures  in  any  of  the 

.0426  given  decimals.     The  third  number  contains  four 

.2140  decimal  places,  and  hence  10000  must  be  a  common 

denominator.      As  annexing  ciphers  to  decimals 

does  not  alter  their  value,  we  give  to  each  number  four  decimal 

places,  by  annexing  ciphers,  and  thus  reduce  the  given  decimals  to 

a  common  denominator. 

Rule.  Give  to  each  numler  the  same  7inmler  of  deci- 
mal places,  hy  anncziiig  ciphers. 

Examples  foe  Practice. 

2.  Reduce  .7,  .073,  .42,  .0020  and  .007  to  a  common  de- 
nominator. 

3.  Reduce  .004,  .00032,  .6,  .37  and  .0314  to  a  common 
denominator. 

4.  Reduce  1  tenth,  46  liundredths,  15  tliousandths,  462 
ten-thousandths,  and  28  hundred-thousandths,  to  a  com- 
mon denominator. 

5.  Reduce  9  thousandths,  9  ten- thousandths,  9  hun- 
dred-thousandths, and  9  miUionths  to  a  common  denomi- 
nator. 

.    6.  Reduce  42.07,  102.006,  7.80,  400.01234  to  a  common 
denominator. 

7.  Reduce  300.3,  8.1003,  14.12614,  210.000009,  and 
1000.02  to  a  common  denominator. 


108  DECIMALS. 

Case  II. 

110.  To  reduce  a  decimal  to  a  common  fraction. 

1.  Reduce  .125  to  an  equivalent  common  fraction. 

OPERATioiT.  Analysis.     Writing  the  decimal  figures 

19^ 105  1^     .125,  over  the  common  denominator  1000,  we 

iitive  YUO^ — F* 

EuLE.  Omit  tlie  decimal  point,  supply  the  proper  de- 
nominator, and  then  reduce  the  fraction  to  its  loivest  terms. 

Examples  for  Practice. 

1.  Eeduce  .08  to  a  common  fraction.  Ans.  ^. 

2.  Reduce  .625  to  a  common  fraction.  Ans.  f. 

3.  Reduce  .375  to  a  common  fraction.  Ans.  f. 

4.  Reduce  .008  to  a  common  fraction.  A7is.  y^. 
6.  Reduce  .4  to  a  common  fraction.  A71S.  f. 
6.  Reduce  .024  to  a  common  fraction.  Ans,  yf^. 

Case  III. 
113.    To  reduce  a  common  fraction  to  a  decimal. 
2.  Reduce  f  to  its  equivalent  decimal. 

OPEEATioiM.  Analysis.      Since  we  can  not 

4)3.0(7  tenthSi  divide  the  numerator  3,  by  4,  re- 

2.8  duce  it  to  tenths  by  annexing  a  ci- 

7~rZI^  ,  w  pher,  and  then  dividing:  we  obtain 

4)20(5   hundreaths.  %  .\^  i  -a  ^    o 

i)r\  "  tenths,  and  a  remainder  of   3 

—  A  -/pj  tenths.     Reducing  this  remainder 

i  \  q  AA  *    *      *  to  hundredths  by  annexing  a  ci- 

'       - — '- pher,  and  dividing  by  4,  we  obtain 

.75,  Ans.  5  hundredths.      The  sum  of  the 

quotients  gives  .75,  the  required  answer. 

Rule.  I.  Annex  ciphers  to  the  7iu7nerator,  and  divide 
hy  the  denominator. 

II.  Point  off  as  many  decimal  places  in  the  result  as  are 
equal  to  the  number  of  ciphers  annexed. 


addition-.  109 

Examples  for  Practice. 

•»  1.  Eeduce  -J  to  a  decimal.  Ans.  .5. 

2.  Reduce  |-  to  a  decimal.  Ajis.  .25. 

3.  Eeduce  |  to  a  decimal  Ans.  .4. 

4.  Reduce  |-  to  a  decimal.  Ans.  .8. 

5.  Reduce  |^  to  a  decimal.  A7is.  .125. 

6.  Reduce  -^  to  a  decimaL  Ans.  .9. 

7.  Reduce  f  to  a  decimal.  ^?25.  .625. 

8.  Reduce  -^V  ^^  ^  decimal.  ^?25.  .04. 

9.  Reduce  j\  to  a  decimal.          -  A7is.  .3125. 

10.  What  decimal  is  equiyalent  to  ^}?  Ans.  .85. 

11.  "What  decimal  is  equivalent  to  -f-^?  Ans.  .1875. 

12.  What  decimal  is  equivalent  to  yf-g-?  Ans.  .016. 


ADDITION. 

114.  Since  the  same  law  of  local  value  extends  both  to 
the  right  and  left  of  units'  place;  that  is, since  decimals  and 
simple  integers  increase  and  decrease  uniformly  by  the  scale 
of  ten, it  is  evident  that  decimals  may  be  added, subtracted, 
multiplied  and  di\'ided  in  the  same  manner  as  integers. 

1.  AYhat  is  the  sum  of  4.314,  36.42, 120.0042,  and  .4276  ? 

OPEKATiON.  Analysis.    Write  the  numbers  so  that  the 

4.314  figures  of  like  orders  of  units  shall  stand  in 

36.42  the  same  columns  ;    that  is,  units  under  units, 

120.0042  tenths  under  tenths,  hundredths  under  hun- 

.4276  dredths,  etc.      This  brings  the  decimal  points 

161  1658  directly  under   each  other.      Commencing  at 

the  right  hand,  add  each  column  separately, 

and  carry  as  in  whole  numbers,  and  in  the  result  place  a  decimal 

point  between  units  and  tenths,  or  directly  under  the  decimal  point 

in  the  numbers  added 


110  DECIMALS. 

KuLE.  I.  Writ^  the  numbers  so  thai  the  decimal  points 
shall  stand  directly  ti?ider  each  other. 

II.  Add  as  in  luhole  numbers,  and  place  the  decimal 'point, 
in  the  result,  directly  under  the  points  in  the  numbers  added. 

Examples  foe  Pkactice. 

2.  What  is  the  sum  of  2.7,  30.84,  75.1,  126.414  and 
3.06?  Ans.  238.114. 

3.  AVhat  is  the  sum  of  1.7,  4.45,  6.75,  1.705,  .50  and 
.05  ?  Ans.  15.155. 

4.  Add  105.7,  19.4,  1119.05,  648.006  and  19.041. 

Ans.  1911.197. 

5.  Add  48.1,  .0481,  4.81,  .00481,  481. 

Ans.  533.96291. 

6.  Add  1.151,  13.29,  116.283,  9.0275  and  .61. 

Ans.  140.3615. 

7.  Add  .8,  .087,  .626,  .8885  and  .49628. 

8.  What  is  the  sum  of  91.003,  16.4691,  160.00471, 
700.05,  900.0006,  .0315  ?  Ans.  1867.55891. 

9.  What  is  the  sum  of  fifty-four,  and  thirty-four  hun- 
dredths ;  one,  and  nine  ten-thousandths ;  three,  and  two 
hundred  seven  millionths ;  twenty-three  thousandths  ; 
eight,  and  nine  tenths ;  four,  and  one  hundred  thirty-five 
thousandths  ?  Ans.  71.399107. 

10.  How  many  acres  of  land  in  four  farms,  containing 
respectively,  61.843  acres,  120.75  acres,  142.4056  acres, 
and  180.750  acres  ?  Ans.  505.7486. 

11.  How  many  yards  of  cloth  in  3  pieces,  the  first  con- 
taining 21 1-  yards,  the  second  36|  yards,  and  the  third 
40.15  yards?  Ans.  98.40. 

12.  A  man  owns  4  city  lots,  containing  32|-,  36|-,  40|, 
42.73  rods  of  land  respectively  ;  how  many  rods  in  all  ? 

Ans.  152.205  rods. 


SUBTEACTIOK. 


Ill 


SUBTRACTIO:^. 
115.    From  124.2750  take  47.3126. 


OPERATIOISr. 

124.2750 
47.3126 


Ans.   76.9624 


Ajstalysis.  Write  the  subtrahend  under 
the  minuend,  placing  units  under  units, 
tenths  under  tenths,  etc.  Commencing  at 
the  right  hand,  subtract  as  in  whole  num- 
bers, and  in  the  remainder  place  the  deci- 
mal point  directly  under  those  in  the  numbers  above.  If  the 
number  of  decimal  places  in  the  minuend  and  subtrahend  are  not 
equal,  they  may  be  reduced  to  the  same  number  of  decimal  places 
before  subtracting,  by  annexing  ciphers. 

EuLE.  I.  Write  the  nuniiers  so  tliat  tlie  decimal  'points 
shall  stand  directly  under  each  other. 

II.  Subtract  as  in  whole  numbers,  and  place  the  decimal 
point  in  the  result  directly  under  the  points  in  the  given 
numbers. 


Examples  for  Peactice. 


(2)                      (3) 

(4) 

Minuend,      12.07                    37.4562 

.003476 

Subtrahend,      4.3264                         .97 

.375 

Remainder,       7.7436                   36.4862 

.628476 

5.  From  463.05  take  17.0613. 

Ans.  445.9887. 

6.  From  134.63  take  101.1409. 

Ans.  33.4891. 

7.  From  189.6145  take  10.151. 

Ans.   179.4635. 

8.  From  671.617  take  116.1. 

Ans.  555.517. 

9.  From  480  take  245.0075. 

Ans.  234.9925. 

10.  Subtract  .09684  from  .145. 

Ans.  .04816. 

11.  Subtract  .2371  from  .2754. 

Ans.  .0383. 

12.  Subtract  215.7  from  271. 

Ans.  55.3. 

13.  Subtract  .0007  from  107. 

Ans.  106.9993. 

14,  Subtract  1.51679  from  27.15. 

Ans.  25.63321. 

112  DECIMALS. 

15.  Subtract  37-|-  from  84.125.  Ans.  46.625. 

16.  Subtract  3|  from  9.3261.  Ans,  5.5761. 

17.  Subtract  25.072  from  112|-.  Afis.  87.553. 

18.  A  man  owned  fifty-four  hundredths  of  a  township 
of  land,  and  sold  fifty-four  thousandths  of  the  same  ;  how 
much  did  he  still  own  ?  Ans.  .486. 

19.  From  10  take  three  millionths.      Ans,  9.999997. 

20.  A  man  owning  475  acres  of  land,  sold  at  different 
times  80.75  acres,  100^  acres,  and  125.625  acres ;  how 
much  land  had  he  left  ?  Ans.  168.5  acres. 


MULTIPLICATIOlSr. 

116.     1.  What  is  the  product  of  .25  multiplied  by  .5. 

OPERATION.  Analysis.    First  multiply  as  in  whole  num- 

,25  bers  ;  then,  since  tlie  multiplicand  has  2  decimal 

.5  places  and  the  multiplier  1,  point  ojff  2  +  1=3 

rrr      .  decimal  places  in  the  product.     The  reason  for 

'  this  will  be  evident,  hy  considering  both  factors 

common  fractions,  and  then  multiplying  as  in  (09),  thus :  .25= j^^^, 

and  .5=  1^0  5  ^^^  t¥o  >^  t^=to¥o*  which  written  decimally  is  .125. 

EuLE.  Multiply  as  in  tvliole  numbers,  and  from  the 
right  hand  of  the  product  poi7it  off  as  many  figures  for 
decimals  as  there  are  decimal  places  in  both  factors. 

1.  If  there  be  not  as  many  flgnres  in  the  product  as  there  are  decimals  in 
both  factors,  supply  the  deficiency  by  prefixing  ciphers. 

2.  To  multiply  a  decimal  by  10,  100, 1000,  remove  the  point  as  many  places  to 
the  right  as  there  are  ciphers  on  the  right  of  the  multiplier. 

Examples  foe  Peaqtice. 

(2)  (3)  (4) 

.241  9.4263  .01346 

.7  .5  .06 


.1687  4.71315  .0008076 


MULTIPLICATIOK. 


113 


5.  Multiply  7.1  by  8.2. 

6.  Multiply  15.5  by  .08. 

7.  Multiply  8.123  by  .09. 

8.  Multiply  4.5  by  .15. 

9.  Multiply  450  by  .02. 

10.  Multiply  341.45  by  .007. 

11.  Multiply  3020  by  .015. 

12.  Multiply  .132  by  .241. 

13.  Multiply  .23  by  .009. 

14.  Multiply  7.02  by  5.27. 

15.  Multiply  .004  by  .04. 

16.  Multiply  2461  by  .0529. 

17.  Multiply  .007853  by  .035. 

18.  Multiply  25.238  by  12.17. 

19.  Multiply  .3272  by  10. 

20.  Multiply  .3272  by  100. 


A71S.  58.22. 

Alls.  1.24. 

Ans.  .73107, 

Ans.  .675. 

Ans.  9. 

Ans.  2.39015. 

Ans.  45.3. 

Ans.  .031812. 

Ans.  .00207. 

Ans.  36.9954. 

Ans.  .00016. 

Ans.  130.1869. 

Ans.  .000274855. 

A71S.  307.14646. 

Ans.  3.272. 

Ans.  32.72. 

Ans.. 327. 2. 

Ans.  1.5. 

Ans.  .00252. 

A71S.  16.432. 


21.  Multiply  .3272  by  1000. 

22.  Find  the  value  of  .25  x  .5  x  12. 

23.  Find  the  value  of  .07  x  2.4  x  .015. 

24.  Find  the  value  of  6^  x  .8  x  3.16. 

25.  If  a  man  travel  3.75  miles  an  hour,  how  far  will  he 
travel  in  9.5  hours?  Ajis.  35.625  miles. 

26.  If  a  sack  of  salt  contain  94.16  pounds,  how  many 
pounds  will  17  such  sacks  contain  ? 

A71S.  1600.72  pounds. 

27.  If  a  man  spend  .87  of  a  dollar  in  1  day,  what  ^vill 
he  spend  in  15.525  days  ?  Ans.  13.50675  dollars. 

28.  One  rod  is  equal  to  16.5  feet ;  how  many  feet  in 
30.005  rods  ?  '  Ans.  495.0825. 

29.  How  many  gallons  of  molasses  in  .54  of  a  barrel, 
there  being  31.5  gallons  in  1  barrel  ? 

Afis.  17.01  gallons. 
S 


114  DECIMALS. 

DIVISION. 

117.    1.  What  is  the  quotient  of  .225  divided  by  .5  ? 

OPERATION.  Analysis.      Perform   the  division  the 

.5  )  .225  same  as  in  whole  numbers,  and  the  only 

.^     J  difficulty  we  meet  with  is  in  pointing  off 

'  *        the  decimal  places  in  the  quotient.     To 

determine  how  many  places  to  point  off,  reduce  the  decimals  to 
common  fractions,  thus ;  .225= j^o^^f^  and  -5= j^^  ;  performing  the  di- 
vision as  in  (97),  we  have  tWo  "^to  =  iWo  ^  ^=iwu  5  ^^^  t'^s  quo- 
tient expressed  decimally,  is  .45.  Here  we  see  that  the  dividend 
contains  as  many  decimal  places  as  are  contained  in  both  divisor 
and  quotient.    Hence  the  foiiomng 

Rule.  Divide  as  in  whole  numbers,  and  from  the  right 
hand  of  the  quotient  ]}Oint  off  as  many  places  for  decimals 
as  the  decimal  places  m  the  dividend  exceed  those  in  the 
divisor. 

1.  If  the  number  of  figures  m  the  quotient  be  less  than  the  exceea  of  the 
decimal  places  in  the  dividend  over  those  in  the  divisor,  the  deficiency  must  be 
supplied  by  prefixing  ciphers. 

2.  If  there  be  a  remainder  after  dividing  the  dividend,  annex  ciphers,  and  con- 
tinue the  division ;  the  ciphers  annexed  are  decimals  of  the  dividend. 

3.  The  dividend  must  always  contain  at  least  as  many  decimal  places  as  the 
divisor,  before  commencing  the  division. 

4.  In  most  business  transactions,  the  division  is  considered  sufficiently  exact 
when  the  quotient  is  carried  to  4  decimal  places,  unless  great  accuracy  is  required. 

5.  To  divide  by  10, 100, 1000,  etc.,  remove  the  decimal  point  us  many  places  to 
the  left  as  there  are  ciphers  on  the  right  hand  of  the  divisor.  » 

Examples  for  Practice. 

(2)  (3)  (4)  (5) 

6 ) .426  .8)3.7624         .05 )  81.60         .009). 00207 

.71.  4.703  1632.  .23 


DIVISIO]^?^.  115 

(b)                                CT  (8) 

,075).9375(12.5      .288)18.0000(.0625  .0025)15.875(6350. 

•  75             '              1728  150 


187               '           720 

87 

150                           576 

75 

375                         1440 

125 

375                         1440 

125 

9.  Di^dde  44  by  .4. 

Ans.  110. 

10.  Divide  15  by  .25. 

Ans.  60. 

11.  Divide  .3276  by  .42. 

Ans.  .78. 

12.  Divide^.00288  by  .08. 

Ans.  .036. 

13.  Divide'.0992  by  .32. 

Ans.  .31. 

14.  Divide  17.6  by  44. 

Ans.  .4. 

15.  Divide  .0000021  by  .0007. 

Ans.  .003. 

16.  Divide  .56  by  1.12. 

Ans.  .5. 

17.  Divide  1496.04  by  10. 

A 

US.   149.604. 

18.  Di\dde  1496.04  by  100. 

A 

ns.   14.9604. 

19.  Divide  1596.04  by  1000. 

A 

ns.  1.59604. 

20.  Divide  4.96  by  100. 

Ans.  .0496. 

21.  Divide  10  by  .1. 

Ans.  100. 

22.  Divide  100  bv  .2. 

Ans.  500. 

23.  If  2.5  acres  produce  34.75  bushels  of  wheat,  how 
much  does  one  acre  produce  ?  Ans.  13.9  bushels. 

24.  If  a  inan  travels  21.4  miles  a  day,  how  many  days 
will  he  require  to  travel  461.03  miles  ? 

25.  If  a  man  build  812.5  rods  of  fence  in  100  days, 
how  many  rods  does  he  build  each  day  ? 

26.  Paid  131.15  dollars  for  61  sheep  ;  what  was  paid  for 
each?  J ws.  2.15  dollars. 


116  decimals. 

Pkomiscuous  Examples. 

1.  Add  twenty-five  hundredths,  six  hundred  fifty-four 
thousandths,  one  hundred  ninety-nine  thousandths,  and 
seven  thousand  five  hundred  sixty-nine  ten- thousandths. 

Ans.  1.8599. 

2.  From  ten  take  ten  thousandths.  A?is.  9.99. 

3.  What  is  the  difference  between  forty  thousand,  and 
forty  thousandths?  Ans.  39999.960. 

4.  Multiply  sixty-five  hundredths,  by  nine  hundredths. 

A}is.  .0585. 

5.  Divide  324  by  6400.  Ans.  .050625. 

6.  Eeduce  .125  to  a  common  fraction.  Ajis.  ^. 
-  7.  Reduce  -J  to  a  decimal  fraction.             'Ans.  .875. 

8.  Divide  .016004  by  .004.  Ans.  4.001. 

9.  Reduce  -|-|-  to  a  decimal  fraction.  Ans.  .68. 

10.  Reduce  .4,  .007,  .1142,  .036,  .00015,  and  .42,  to  a 
common  denominator. 

11.  At  13.9  dollars  a  ton,  what  will  2.5  tons  of  hay  cost  ? 

A?is.  34.75  dollars. 

12.  If  a  pound  of  sugar  cost  .09  dollars,  hoAV  many 
pounds  can  be  bought  for  5.85  dollars?  Ans.  65  pounds. 

13.  If  40.02  bushels  of  potatoes  are  raised  upon  1  acre 
of  land,  how  many  acres  will  be  required  to  raise  4580.64 
bushels?  Ans.  114.458  acres. 

14.  At  11  dollars  a  ton,  how  much  hay  can  be  bought 
for  13.75  dollars?      •  A7is.  1.25  tons. 

15.  If  a  man  travel  32.445  miles  in  a  day,  how  far  can 
he  travel  in  .625  of  a  day  ?  Ans.  20.278125  miles. 

16.  If  2  pounds  of  sugar  cost  .1875  dollars,  what  will 
be  the  cost  of  10  pounds?  A7is.  .9375  dollars. 

17.  If  3  barrels  apples  cost  19.125  dollars,  what  will  be 
the  cost  of  100  barrels  ?  Ans.  637.5  dollars. 


UNITED     STATES     MONET.  117 

uj^ited  states  mojstey. 

118.  United  States  Money  is  the  legal  currency 
of  the  United  States,  and  was  established  by  act  of  Con- 
gress, August  8,  1786.  Its  denominations  and  their  rela- 
tive values  are  shown  in  the  following 

Table. 

10  mills  (711.)  make  1  cent ct. 

10  cents  ^'     1  dime d. 

10  dimes  ''     1  dollar 8. 

10  dollars  ''      1  eagle E. 

The  currency  of  the  United  States  is  decimal  currency,  and  is  sometimes 
called  Federal  Money, 

119.  The  character  S,  before  any  number,  indicates 
that  it  expresses  United  States  money.  Thus,  $75  ex- 
presses 75  dollars. 

120.  The  dollar  is  the  miit  of  United  States 
money  ;  dimes,  cents,  and  mills  are  fractions  of  a  dollar, 
and  are  separated  from  the  doWav  hj  the  decimal  pomt  {.); 
thus,  two  dollars  one  dime  two  cents  five  mills  are  written, 

l>/^.l/iO. 

131.   By  examining  the  above  table  we  find, 

1st.  That  the  dollar  being  the  unit,  dimes,  cents,  and 
mills  are  respectively  tenths,  hundredths,  and  thousandths 
of  a  dollar. 

2d.  That  the  denominations  of  United  States  money 
increase  and  decrease  the  same  as  simple  numbers  and 
decimals,  and  are  expressed  according  to  the  decimal 
system  of  notation. 

Hence  we  conclude  that 

United  States  money  may  le  added,  siiltracted,  multi- 
plied and  divided  in  the  same  manner  as  decimals. 


118  UKITED     STATES     MON'EY. 

Dimes  are  not  read  as  dimes,  but  the  two  places  of  dimes 
and  cents  are  appropriated  to  cents  ;  thus  1  dollar  3  dimes  2 
cents,  or  $1.32,  are  read  one  dollar  tMrty-two  cents;  hence, 

When  the  number  of  cents  is  less  than  10,  we  write  a 
ciplier  tefore  it  in  the  jjlace  of  dimes. 

The  half  cent  is  frequently  written  as  5  mills ;  thus,  24i  cents,  is  written 
$.245. 

Examples  for  Practice. 

1.  Write  five  dollars  twenty-five  cents.    Ans.  $5.25. 

2.  Write  four  dollars  eight  cents.  Ans.  $4.08. 

3.  Write  twelve  dollars  thirty-six  cents. 

4.  Write  seven  dollars  sixteen  cents. 

5.  Write  ten  dollars  ten  cents. 

6.  .Write  sixty-five  cents  four  mills.         Ans.  $.654. 

7.  Write  one  dollar  five  cents  eight  mills.      $1,058. 

8.  Write  eighty-seven  cents  five  mills.     Ans.  $.875. 

9.  Write  one  hundred  dollars  one  cent  one  mill. 

Ans.  $100,011, 
10.  Bead  $4.07  ;  $3,094  ;  $10.50 ;  $25.02. 

REDUCTION. 

122,     1.  How  many  cents  are  there  in  75  dollars  ? 

OPERATION.  Analysis.     Since  in    1    dollar    there    are 

75  100  cents,  in  75  dollars  there  are   75  times 

2Q()  100   cents  or  7500  cents.     To    multiply   by 

10,    100,    etc ,    annex    as    many    ciphers    to 

VOUU  cents,     j^^q  multiplicand  as  there  are  ciphers  in  the 
multiplier,  (62).    Hence 

To  cliange  dollars  to  cents,  multiply  hy  100 ;  that  is, 
annex  two  ciphers.     And 

To  change  dollars  to  mills,  annex  three  ciphers. 
To  change  cejits  to  mills,  annex  oke  cipher. 


reductiois'.  119 

Examples  for  Practice. 

2.  Reduce  124  to  cents.  Ans.  2400  cents. 

3.  Eednce  $42  to  cents.  A71S,  4200  cents. 

4.  Eeduce  $14  to  mills.  Ans.  14000  mills. 

5.  Eeduce  $102  to  cents. 

6.  Change  $35  to  mills. 

7.  Change  6Q  cents  to  mills.  Ans.  G60  mills. 

8.  Change  73  cents  to  mills. 

To  change  dollars  and  cents,  or  dollars,  cents,  and  mills  to  mills,  remove  tha 
decimal  point  and  sign,  $. 

9.  Change  $4.28  to  cents.  Ans.  428  cents. 

10.  Change  818.07  to  cents.  Ans.  1807  cents. 

11.  Chanjre  86.325  to  mills.  Ans.  6325  mills. 

12.  In  87.01  how  many  cents? 

13.  In  94  cents  how  many  mills  ? 

14.  In  $51  how  many  cents  ? 

1.  In  3427  cents  how  many  dollars? 

OPERATION.  Analysis.    Since  100  cents  equal 

1|00  )  34127  1   (iollar,   3427  cents  equal  as  many 

rrTTT      .  dollars  as  100  is  contained  times  in 

$34.27,  Ans.       g^^?,  wliich  is  3427  times.    To  divide 

by  10,  100,  etc.,  cut  off  as  many  figures  from  the  right  of  the 

dividend  as  there  are  ciphers  in  the  divisor,  (72).     Hence 

To  change  cents  to  dollars,  divide  ly  100 ;  that  is,  point 
off  TWO  figures  from  the  right.     And 

To  change  mills  to  dollars,  point  off  tb:reb  figures. 
To  chayige  mills  to  cents,  point  off  o^e  figure. 

Examples  for  Peactice. 

2.  Change  972  cents  to  dollars.  Ans.  $9.72. 

3.  Change  1609  cents  to  dollars.  Ans.  816.09. 

4.  Change  3476  mills  to  dollars.  Ans.  $3,476, 


120  UKITED     STATES     MOKET. 

5.  In  34:6^1  cents  how  many  dollars? 

6.  In  10307  cents  how  many  dollars  ? 

7.  In  203062  mills  how  many  dollars?  Ans.  1203.062. 

8.  Eeduce  672  mills  to  cents.  Ans,  $.672. 

9.  Reduce  3104  mills  to  dollars. 
10.  Reduce  17826  cents  to  dollars. 

ADDITION. 

123.     1.  What  is  the  sum  of  $12.50,  $8,125,  $4,076, 

$15,375  and  $22? 

OPERATION. 
$12.50 

g  ]^25  Analysis,     Writing  dollars  under  dol- 

4.  Qi^g  lars,  cents  under  cents,  etc.,  so  tliat  the 

15.375  decimal  points   shall  stand    under    each 

22. 000  other,  we  add  and  point  ofiF  as  in  addition 

M.'ZT'ZZ^     .  of  decimals. 

$62,076,  ^?i5. 

Rule.  I.  Write  dollars  under  dollars^cents  under  cejits,  etc. 

IL  Add  as  in  simple  numbers,  and  place  the  point  in  the 

mnount  as  in  addition  of  decimals. 

Examples  for  Practice. 


(2) 

(3) 

W 

(5) 

$126,085 

$100,375. 

$750.00 

$1042.875 

42.64 

13.09 

140.07 

427.035 

304.127 

65.82 

35.178 

50.50 

•  14.42 

400.00 

6.004 

7.08 

6.  What  is  the  sum  of  30  dollars  9  cents ;  200  dollars 
63  cents;  27  dollars  36  cents  4  mills,  and  10  dollars 
16  cents?  Ans.  $268,244. 

7.  Add  390  dollars  37  cents  5  mills,  187  dollars  50 
cents,  90  dollars  5  cents  5  mills,  and  400  dollars  40  cents. 

Ans.  $1068.33. 


ADDITION.  121 

8.  A  lady  paid  845.40  for  some  furs,  112.375  for  a  dress, 
$5  for  a  bonnet  and  $1,125  for  a  pair  of  gloves ;  what  did 
she  pay  for  all  ?    • 

9.  A  farmer  sold  a  cow  for  120,  a  horse  for  $96.50,  a 
yoke  of  oxen  for  $66,875,  and  a  ton  of  hay  for  $9.40  ; 
what  did  he  receive  for  all  ?  A7is.  $192,775. 

10.  Bought  a  hat  for  $4.50,  a  pair  of  boots  for  $5.62-|-, 
an  umbrella  for  $2.12^,  and  a  pair  of  gloves  for  $.87^ ; 
what  was  the  cost  of  the  whole  ?  A71S.  $13,125. 

11.  A  grocer  bought  a  barrel  of  sugar  for  $17.84,  a  box 
of  tea  for  $36.12|-,  a  cheese  for  $4,  and  a  tub  of  butter  for 
$7.09  ;  what  was  the  cost  of  all? 

12.  A  merchant  bought  a  quantity  of  goods  for  $458.25, 
paid  for  duties  $45  ;  for  freights  $98.62|-,  and  for  insur- 
ance $16.40  ;  what  was  the  whole  cost  ? 

Ans.  $618,275. 

13.  Bought  some  sugar  for  $1.75,  some  tea  for  $.90, 
some  butter  for  $2.12J,  some  eggs  for  $.37|-,  and  some  spice 
for  $.25  ;  what  was  the  cost  of  the  whole  ?   Ans.  $5.40. 

14.  Paid  for  building  a  house  $1045.75,  for  painting  the 
same  $275.60,  tor  furniture  $648.87|-,  and  for  carpets 
$105.10  ;  what  was  the  cost  of  the  house  and  furnishing  ? 

A71S.  $2075.325. 

15.  A  farmer  receives  120  dollars  45  cents  for  wheat, 
36  dollars  62 J  cents  for  corn,  14  dollars  9  cents  for  pota- 
toes, and  63  dollars  for  oats ;  how  much  does  he  receive 
for  the  whole  ? 

16.  A  lady  who  went  shopping,  bought  a  dress  for  7  dol- 
lars 27  cents,  trimmings  for  87-|-  cents,  some  tape  for 
6  cents,  some  thread  for  12  J-  cents,  and  some  needles  for 
9  cents  ;  what  did  she  pay  for  all  ?  A71S.  $8.42. 


122  UNITED     STATES     MOi?^EY. 

SUBTEACTION, 

124.  1.  From  246  dollars  82  cents  5  mills,  take  175 
dollars  27  cents. 

OPERATioisr.  Analysis.     Writing  tlie  less  number 

1246.825  under  the  greater,  dollars  under  dollars, 

175.27  cents  under  cents,  etc. ,  we  subtract  and 

T77~r^    J  point  off  in  the  result  as  in  subtraction  of 

^              '  decimals. 

Rule.  I.  Write  the  siihtraliend  under  the  mimmid, 
dollars  under  dollars,  cents  imder  cents,  etc, 

II.  Subtract  as  in  simple  nuvfibers,  and  place  the  point 
in  the  remainder  as  in  subtraction  of  decimals. 

Examples  poe  Peactice. 
(2.) 

From    $125.05 
Take  43.278 

Ans.    $81,772      $227,015        $111,125        $41,315 

6.  From  $3472.50  take  $1042.125.     Ans.  $2430.375. 

7.  From  $540  take  $256.67.  Ans.  $283.33. 

8.  From  $82.04  take  $80,625.  Ans.  $1,415. 

9.  From  3  dollars  10  cents,  take  75  cents.  Ans.  $2.35. 

10.  From  10  dollars,  take  5  dollars  10  cts.   Ans.  $4.90. 

11.  From  100  dollars,  take  50  dollars  50  cents. 

12.  From  1001  dollars  9  cents,  take  300  dollars. 

13.  From  2  dollars  take  75  cents.  Ans.  $1.25. 

14.  From  96  cents  take  12-|^  cents.  Ans.  $.835. 

15.  From  1  dollar  take  25  cents.  Ans.  $.75. 

16.  From  50  cents  take  37  cents  5  mills.  Ans.  $.125. 

17.  From  5  dollars  take  50  cents  8  mills.  Ans.  $4,492. 

18.  From  4  dollars  take  1  dollar  40  cents  5  mills. 

19.  Sold  a  korse  for  $200,  which  was  $45.50  more  than 
he  cost  me  ;  what  did  he  cost  me?  Ans.  $154.50. 


(3.) 

(4.) 

(5.) 

$327,105 

$112,000 

$43,375 

100.09 

.875 

2.06 

SUBTRACTIO^q".  123 

20.  A  man  bought  a  farm  for  $4640,  and  sold  it  for 
$5027.50  ;  what  did  he  gain  ?  Ans.  $387.50. 

21.  Borrowed  825  and  returned  $15. 60  ;  how  much  re- 
mained unpaid  ?  A7is.  $9.40. 

22.  A  merchant  having  $10475,  paid  $2426  for  a  store, 
And  $5327.875  for  goods  ;  how  much  money  had  he  left  ? 

A71S.  $2721.125. 

23.  Bought  a  sack  of  flour  for  $3.12|- ;  how  much 
change  must  I  receive  for  a  5  dollar  bill?    A?is.  $1,875. 

24.  Bought  groceries  to  the  amount  of  $1,875  ;  how 
much  change  must  I  receive  for  a  2  dollar  bill  ? 

A)is.  12 J-  cents. 

25.  Paid  $375  for  a  pair  of  horses,  and  sold  one  of  them 
for  $215.50 ;  what  did  the  other  one  cost  me  ? 

A71S.  $159.50. 

26.  I  started  on  a  journey  with  $50  and  paid  $10.62|- 
Railroad  fare,  $7.38  stage  fare,  $5.96  for  board  and  lodg- 
ing, and  $.75  for  porterage  ;  how  much  money  had  I  left  ? 

A?is.  $25,285. 

27.  A  farmer  sold  some  wool  for  $27.16,  and  a  ton  of 
hay  for  $14.80.  He  received  in  payment  a  barrel  of  flour 
worth  $6,875,  and  the  remainder  in  money  ;  how  much 
money  did  he  receive?  Ajis.  $35,085. 

28.  A  woman  sold  a  grocer  some  butter  for  $1.48,  and 
some  eggs  for  $.94.  She  received  a  gallon  of  molasses 
worth  40  cents,  a  pound  of  tea  worth  75  cents,  and  a 
pound  of  starch  worth  12|-  cents  ;  how^  much  is  still  her 
due  ?  Ans.  $1,145. 

29.  A  tailor  bought  a  piece  of  broadcloth  for  $87.50, 
and  a  piece  of  cassimere  for  $62.75.  He  sold  both  pieces 
for  $170.87i  ;  what  did  he  gain  on  both? 

A71S.  $20,625. 


134  UKITED     STATES     MOKET. 

MULTIPLICATIOIT. 
125.     1.  Multiply  $26,145  by  34. 

OPERATION. 

$26,145  Analysis.      Multiply  as   in    simple 

34  numbers,  always  regarding  the  multi- 

plier as  an  abstract  number,  and  point 
off  from  the  right  hand  of  the  result,  as 
in  multiplication  of  decimals. 


104580 
78435 


$888,930,  Ans, 

EuLE.    Multiply  as  in  simple  7iuml)ers,  and  place  the 
point  ill  the  product  as  in  multiplication  of  decimals. 


Examples  foe  Practice. 

(2.) 

$327.48 
15 

(3.)                 (4.) 
$82,375          $160.09 
46                  87 

(5.) 

$97,875 
123 

6.  What  cost  8  cords  of  wood,  at  $3.50?    Ans.  $28. 

7.  What  cost  14  barrels  of  flour,  at  $5.85  a  barrel  ? 

8.  What  cost  25  bushels  of  corn,  at  75  cents  a  bushel  ? 

9.  At  $2,125  a  yard,  what  will  18  yards  of  silk  cost? 

10.  At  $.875  apiece,  what  will  be  the  cost  of  9  turkeys? 

11.  A  farmer  sold  40  bushels  of  potatoes  at  37|-  cents  a 
bushel,  and  21  barrels  of  ajpples  at  $2.25  a  barrel ;  what 
did  he  receive  for  both  ?  Ans.  $62.25. 

12.  Bought  124  acres  of  land  at  $35.75  an  acre,  and 
sold  the  whole  for  $6000  ;  what  did  I  gain  ? 

A71S.  $1567. 

13.  What  will  be  the  cost  of  275  bushels  of  oats,  at  42 
cents  a  bushel  ?  Ans.  %llb.bO. 

14.  A  grocer  bought  160  pounds  of  butter,  at  14  cents  a 
pound,  and  paid  25  pounds  of  tea,  worth  56  cents  a  pound, 
and  the  remainder  in  cash  ;  how  much  money  did  he  pay  ? 


DIVISIONS".  125 

15.  What  will  be  the  cost  of  15  yards  of  broadcloth,  at 
$4.87iayard?  A^is.  873.125. 

16.  A  grocer  bought  a  tub  of  butter  containing  84 
pounds,  at  12-|-  cents  a  pound,  and  sold  the  same  at  15 
cents  a  pound  ;  what  did  he  gain?  Ans.  $2.10. 

17.  A  farmer  took  3  tons  of  hay  to  market,  for  which 
he  received  $9.38  a  ton.  He  bought  2  barrels  of  flour,  at 
$6.94  a  barrel,  and  12  pounds  of  tea,  at  $.625  a  pound ; 
how  much  money  had  he  left  ?  Ans.  $6.76. 

DIVISIOI^. 
126.     1.  Di^dde  $136  by  64. 

OPERATIOISr. 

64 )  $136,000  ( $2,125,  Ans. 
128 

Q  Analysis.    Divide  as  in  simple 

^  .  numbers,  and  as  there  is  a  remaiii- 

. der  after  dividing  the  dollars,  re- 

160  duce  the  dividend  to  mills,   by 

128  annexing  three  ciphers,  and  con- 

qorj  tinue  the  division. 

320 

Rule.  Divide  as  in  simple  number s^  and  place  the 
point  in  the  quotient,  as  in  division  of  decimals. 

In  business  transactions  it  is  never  necessary  to  carry  the  division  further 
than  to  mills  in  the  quotient. 

Examples  for  Peactice, 

(2.)  (3.)  (4.)  (5.) 

5 )  $43.50        10)  $36.00        8)  $371  12)  $169.50 

$8.70  $3.60  $46,375  $14,125 


126  UNITED     STATES     MONEY. 

6.  Divide  $13.75  by  11.  Ans.  $1.25. 

7.  Divide  $162  by  36.  Ans.  $4.50. 

8.  Divide  $246.30  by  15.  Ans.  $16.42. 

9.  Divide  $1305  by  18.  Ans.  $72.50. 

10.  Divide  $2.25  by  9.  Ans.  $.25. 

11.  Divide  $658  by  280.  A7is.  $2.35. 

12.  Divide  $195.75  by  29.  Ans.  $6.75. 

13.  Di\'ide  $1388  by  100.  Aois.  $13.88. 

14.  Divide  $2675.75  by  278.  Ans.  $9,625. 

15.  Divide  $68  by  32.  Ans.  $2,125. 

16.  Paid  $168.48  for  144  bushels  of  wheat  ;   what  was 
the  price  per  bushel  ?  Ans.  $1.17. 

17.  Paid  $2.80  for  35  pounds  of  sugar  ;  what  was  the 
price  per  pound  ?  Ans.  $.08. 

18.  If  54  cords  of  wood  cost  $135,  what  is  the  price  per 
cord?  Ans.  $2.50. 

19.  Bought  125  bushels  of  oats  for  $62.50  ;  what  was 
the  cost  per  bushel?  Ans.  $.50. 

20.  If  70  barrels  of  apples  cost  $175,  what  will  1  barrel 
cost  ?  A71S.  $2.50. 

21.  If  100  acres  of  land  cost  $3156.50,  what  will  be  the 
cost  of  1  acre  ?  Ans.  $31,565. 

22.  Paid  $148.75  for  170  bushels  of  barley  ;  what  was 
the  cost  per  bushel  ?  Ans.  $.875. 

23.  If  13  pounds  of  tea  cost  $9.88,  what  will  1  pound 
cost  ? 

24.  Bought  2500  pounds  of  butter  for  $625  ;  what  was 
the  cost  per  pound  ?  Ans.  25  cents. 

25.  Bought  2450  pounds  of  pork  for  $153.12J- ;  what 
was  the  cost  per  pound  ?  Ans.  6|-  cents. 

26.  Bought  4  barrels  of  sugar,  each  containing  200 
pounds,  for  $72  ;  what  was  the  cost  per  pound  ? 


PROMISCUOUS     EXAMPLES.  127 


Promiscuous  Examples. 

1.  A  mercliant  bought  14  boxes  of  tea  for  8560  ;  but  it 
being  damaged,  lie  was  obliged  to  sell  it  for  $106.75  less 
than  it  cost  him  ;  what  did  he  receive  per  box  ? 

Ans.  $32,375. 

2.  A  farmer  sold  120  bushels  of  wheat,  at  $1.12|-  a 
bushel,  and  received  in  payment  27  barrels  of  flour  ;  what 
did  the  flour  cost  him  per  barrel  ? 

3.  If  35  yards  of  cloth  cost  $122.50,  what  will  29  yards 
cost?  Ans.  $101.50. 

4.  If  4  tons  of  coal  cost  $35.50,  what  will  12  tons  cost? 

Ans.  1106.50. 

5.  If  29  pounds  of  sugar  cost  $3,625,  what  will  15 
pounds  cost  ?  Ans.  $1,875. 

6.  If  12  barrels  of  flour  cost  $108,  what  will  18  barrels 
cost  ?  •  A71S.  $162. 

7.  If  3  bushels  of  wheat  cost  $4.35,  what  will  30  bushels 
cost?  Ans.  $43.50. 

8.  A  man  bought  a  farm  containing  125  acres,  for 
$2922.50  ;  for  what  must  he  sell  it  per  acre  to  gain  $500  ? 

Ans.  $27.38. 

9.  A  farmer  exchanged  50  bushels  of  corn  worth  70 
cents  a  bushel,  for  28  bushels  of  wheat ;  what  was  the 
wheat  worth  a  bushel  ?  Ans.  $1.25. 

10.  A^erson  ha^dng  $15000,  bought  30  bales  of  cotton, 
each  bale  containing  940  pounds,  at  10  cents  a  pound;  he 
next  paid  $6680  for  a  house,  and  then  bought  1000  barrels 
of  flour  with  what  money  he  had  left  ;  what  did  the  flour 
cost  him  per  barrel  ?  Ans.  $5.50. 

For  a  fnll  and  complete  development  and  application  of  Decimals  and  United 
States  money,  the  pupil  is  referred  to  the  Author's  Progressive  Practical,  and 
Higher  Arithmetics, 


128  UNITED     STATES     MONEY. 

BILLS. 

127.  A  Bill,  in  business  transactions,  is  a  written 
statement  of  articles  bought  or  sold,  together  with  the 
prices  of  each,  and  the  whole  cost. 

Find  the  cost  of  the  several  articles,  and  the  amount  or 
footing  of  the  following  bills  : 

(1.) 

Chicago,  Sept.  20,  1871. 

Mr.  J.  C.  Smith, 

BoH  of  Silas  Johnsok, 

36  pounds  sugar  at  8  cents  a  pound,  12.88 

18  pounds  coffee  at  15  cents  a  pound,  2.70 

24  pounds  butter  at  IJ  cents  a  pound,  4,32 
10  dozen  eggs  at  12-|-  cents  a  dozen,  1.25 

4  gallons  molasses  at  44  cents  a  gallon,  1.76 

Ans,  $12.91 

(2.) 

EocHESTER,  Jan.  25,  1872. 
John  Dabney,  Esq., 

BoH  of  Bardwell  &  Co., 
14  pounds  coffee  sugar  at  11  cents  a  pound,        11.54 

6  pounds  Y.  H.  tea  at  62J  cents  a  pound,  3.75 

25  pounds  No.  1  mackerel  at  6  cents  a  pound,     1.50 

5  bushels  potatoes  at  37J  cents  a  bushel,  *  1.875 
3  gallons  syrup  at  80  cents  a  gallon,  2.40 

7  dozen  eggs  at  16  cents  a  dozen,  1.12 

Ans.  $12,185 
Received  Payment, 

Bardwell  &  Co., 

per  Adams. 


BILLS.  129 

(3.) 

Memphis,  Aug.  20,  1872. 
Mr.  S.  P.  Haile, 

BoH  of  Patterson"  &  Co., 

20  chests  Green  Tea  at $22.50 

16       ''     Black      ''    '',.., 18.75 

14  ''     Imperial^^    " 32.87i 

15  sacks  Java  Coffee    '' 17.38 

25  boxes  Oranges        ^' 4.62|-    

$1586.575 
Received  Payment^ 

Pattersoi^  &  Co. 

(4.) 

Oswego,  Sept.  4,  1871. 

James  Coroval  &  Co., 

BoH  of  Collins  &  Sok, 

12  yards  Broadcloth     at $3.84 

18      ''  Cassimere       '' 2.25 

10      "  Satinet  '' 87^ 

42      ''  Flannel  ''  ..., 45 

35      *'  Black  Silk     /' 1.18 

$155.53 
(5.) 

Boston-,  April  10,  1872. 

J.  G.  Bennet  &  Son, 

BoH  of  Butler,  King  &  Co., 

14  Plows  at $10.50 

8  Harrows      '' 9.80 

120  Shovels        '' 90 

175  Hoes  " 62J 

$442,775 


130  OOMPOUKD     NUMBEES. 

COMPOUS'D  NUMBEES. 

138.  A  Simple  Number  is  either  an  abstract  number, 
or  a  concrete  number  of  but  one  denomination.  Thus, 
48,  926  ;  48  dollars,  926  miles. 

139,  A  Compooiul  Number  is  a  concrete  number 
whose  yalue  is  expressed  in  two  or  more  different  denomi- 
nations. Thus,  32  dollars  15  cents ;  15  days  4  hours  25 
minutes. 

130.  A  Scale  is  a  series  of  numbers,  descending  o:: 
ascending,  used  in  operations  upon  numbers. 

In  simple  numbers  and  decimals  the  scale  is  uniformly  10 ;  in  compound  rmny 
bers  the  scales  are  varying. 

CURRElSrCY. 

1.  United  States  Mon^ey. 

131.  The  currency  of  the  United  States  is  decimal  cur- 
rency, and  is  sometimes  called  Federal  Honey. 

Table. 

10  Mills  (m.)  make  1  Cent ct. 

10  Cents  "       1  Dime d. 

10  Dimes  "      1  Dollar $. 

10  Dollars         "      1  Eagle E. 

Unit  Equivalents. 

ct.  m. 

d.  1  =        10 

$  1  =      10  =      100 

E.         1  =     10  =     100  =     1000 

1  =r  10  =  100  =  1000  =  10000 

Coins.  Tlie  gold  coins  are  the  double  eagle,  eagle,  half- 
eagle,  quarter-eagle,  the  three-dollar,  and  one-dollar  pieces. 

The  silver  coins  are  the  dollar,  the  half-dollar,  quarter- 
dollar,  the  twenty-cent,  and  ten-cent  pieces. 

The  nickel  coins  are  the  five-cent,  and  three-cent  pieces. 
^    The  bronze  coins  are  the  one-cent  pieces. 


MOKEY     Ai^D     CUKKElfCIES.  131 

II.    CAiq^ADA   MOi^^ET. 

132.  The  currency  of  the  I>ominioii  of  Canada  is 
decimal,  and  the  table  and  denominations  are  the  same  as 
those  of  United  States  money. 

The  currency  of  the  Dotninion  of  Canada  was  made  uniform  July  1st,  1871. 
Before  the  adoption  of  the  decimal  system,  pounds,  shillings,  and  pence 
were  used. 

CoiKS.  The  new  Canadian  coins  are  silver  and  bronze. 
The  silver  coins  are  50-cent  piece,  25-cent  piece,  10-cent 
piece,  and  5-cent  piece. 
The  Ironze  coin  is  the  cent. 

The  gold  coin  used  in  Canada  is  the  Bntish  Sovereign^  worth  $4.86f ,  and  the 
EcUf- Sovereign. 

III.  English  Mokey. 

133.  English  or  Sterling  money  is  the  currency  of 
Great  Britain. 

Table. 

4  Farthings  (far.  or  qr.)  make  1  Penny d. 

13  Pence  "      1  ShiUing s. 

20  Shillings  **      1  Pound  or  Sovereign £  or  so  v. 

Unit  Equivalents. 

d.         far. 
s.  1=4 

£         1  =    12  =    48 
1  =  20  =  240  =  960 

Scale — ascending,  4,  12,  20  ;  descending,  20,  12,  4. 

Farthings  are  generally  expressed  as  fractions  of  a  penny  ; 
sometimes  called  1  quarter,  (qr.)  =  ^d, ;  3  far.  =  f  d. 

Coins.  The  gold  coins  are  the  sovereign  (=£1)  and 
the  half-sovereign  (rzzlOs.). 

The  silver  coins  are  the  crown,  half-crown,  florin,  shil- 
ling, sixpenny,  fourpenny,  and  threepenny  pieces. 

The  cojp^er  coins  are  the  penny,  haK-penny,  and  farthing. 


13^  COMPOUND     NUMBEES. 


WEIGHTS. 

134.  Weight  is  a  measure  of  the  quantity  of  matter  a 
body  contains,  determined  according  to  some  fixed  standard. 

I.  Trot  Weight. 

135.  Troy  Weight  is  used  in  weighing  gold,  silver, 
and  jewels  ;  in  philosophical  experiments,  etc. 

Table. 

24  Grains  (gr.)  make  1  Pennyweight pwt.  or  dwt. 

20  Pennyweights  "      1  Ounce oz. 

12  Ounces  "     1  Pound lb. 

Unit  Equivalents. 

pwt.        gr. 

oz.  1  =      24 

lb.        1  =    20  =    480 

1  =  12  =  240  =  5760 

Scale — ascending,  24,  20,  12  ;  descending,  12,  20,  24. 

II.  Ayoikdupois  Weight. 

136.  Avoirdupois  Weight  is  used  for  all  the  ordinary 
purposes  of  weighing. 

Table. 

16  Ounces  (oz.)    make  1  Pound lb. 

100  Pounds               ''      1  Hundred- weight.. ..  .cwt. 
20  Cwt.,  =  2000  lbs.,     1  Ton T. 

Unit  Equivalents. 

lb.  oz. 

cwt.  1  =        16 

T.         1  =    100  =    1600 

1  =  20  =  2000  =  32000 

Scale— ascending,  16,  100,  20  ;  descending.  20,  100, 16. 


WEIGHTS.  13S 

The  long  or  gross  ton,  hundred-weight,  and  quarter  were  formerly  in  common 
use  •  but  they  are  now  seldom  used  except  in  estimating  English  goods  at  the 
U.  S'.  Custom-House,  and  in  freighting  and  wholesaling  coal  from  the  Pennsyl- 
vania mines. 

Long  Tok  Table. 

28  Pounds  make  1  Quarter qr. 

4  Quarters  =  112  lb.       "       1  Hundred- weight cwtc 

20  Cwt.  =  2240  lb.  "       1  Ton T. 

« 

The  following  denominations  are  also  in  use : 

56  Pounds  make  1  Firkin  of  butter. 
196 
200 
280 

32 

48 

56 

60 


"  1  Barrel  of  flour. 

"  1       "       "  beef,  pork,  or  fish. 

"  1  Bushel  "  salt  at  the  N.  Y.  State  salt  works. 

"  1       "       "  oats. 

"  1       "       "  barley. 

**  1       '*       "  corn  or  rye. 

"1       "       "  wheat. 

III.  Apothecaeies'  Weight. 

137.  Apothecaries'  Weight  is  used  by  apothecaries 
and  physicians  in  compounding  medicines  ;  but  medicines 
are  bought  and  sold  by  Avoirdupois  Weight. 

Table. 

20  Grains  (gr.)  make  1  Scruple  sc.  or  3. 

3  Scruples  ' '      1  Dram dr.  or  3  . 

8  Drams  "      1  Ounce oz.  or  §  . 

12  Ounces  "      1  Pound lb.  or  lb. 

Unit  Equivalents. 

sc.  gr. 

dr.         1  =      20 

oz.         1  =      3  =      60 

lb.         1  =    8  =     24  =    480 

1  =  12  =  96  =  288  =  5760 

Scale — ascending,  20,  3,  8, 12  ;  descending,  12,  8,  3,  20. 


134  COMPOUND     KUMBEKS. 

138.  COMPARATIYE   TaBLE   OF   WEIGHTS. 

Troy.  Avoirdupois.  Apothecaries. 

1  pound  =  5760  grains,     =  7000  grains,  =  5760  grains. 

1  ounce    =480       "  =     437.5     "  =480       " 

175  pounds,  =     144  pounds,  =     175  pounds. 

MEASURES    OF    EXTENSION. 

139.  Extension  has  three  dimensions — length,  IreadtJi, 

and  thich7iess, 

A  Line  has  only  one  dimension — length. 

A  Surface  or  Area  has  two  dimensions — length  and 
breadth. 

A  Solid  or  Body  has  three  dimensions — length,  breadth, 
and  thickness. 

\ 
\ 

I.  LoJTG  Measure. 

140.  Long-  Measure,  also  called  Lineal  Measure,  is 
used  in  measuring  lines  or  distances. 

Table. 

12     Inches  (in.)  make  1  Foot ft. 

3    Feet  "      1  Yard yd. 

^  Yards,  or  16|-  ft.,      "     1  Rod rd. 

330    Rods  "     1  Statute  Mile mi. 

Unit  Equivalents. 

ft.  in. 

yd.              1     =  12 

rd.             1     =        3     =  36 

mi.          1  =        5i  =      16|  =  198 

1  =  320  =  1760'   =  5280     =  63360 

Scale — ascending,  12,  3,  5|,  320  ;  descending,  320,  5^,  3,  12. 


MEASURES     OF     EXTEN"SIOK.  135 

141.  Tlie  following  denominations  are  also  in  use  : — 

3  Barleycorns  make  1  Inch,  used  by  shoemakers. 

4  Inches  "      1  Hand,  \  ^^^^  *^  measure  the  height  of 

(  horses. 
9  *'  "      1  Span. 

21.888     "  "      1  Sacred  cubic. 

3        Feet  "      1  Pace. 

g  «  "I  Fathom,  \  ^^^^    ^^  measuring  depths 

(  at  sea, 

1.15  Statute  miles  "      1  Geographic  mile,   \  f  ^^  ^^  measuring 

(  distances  at  sea. 
3    Geographic"      "       1  League. 
60  "  "or      I  ^  Decree    \  °^  latitude  on  a  meridian,  or 

69.16  Statute      "      "    f  ^   '  '   ( of  longitude  on  the  equator. 

360      Degrees  "       the  Circumference  of  the  earth. 

For  the  pui-pose  of  measuring  cloth  and  other  goods  sold  by  the  yard,  the 
yard  is  divided  into  halves,  quarters,  fourths,  eighths^  and  sixteenths.  The  old 
table  of  Cloth  Measure  is  practically  obsolete. 

Surveyors'  Lokg  Measure. 

142.  A  Gunter's  Chain,  used  by  land  surreyors,  is  4 
rods  or  66  feet  long,  and  consists  of  100  links. 

Table. 

7.92  Inches  (in.)  make  1  Link I. 

25      Links  "      1  Rod rd. 

4      Rods,  or  66  feet,  "      1  Chain ch. 

80      Chains  "      1  Mile mi. 

Unit  Equitalents. 

1.  in. 

rd.  1  =      7.92 

ch.  1  =     25  =      198 

mi.        1  =      4  =   100  =      792 

1  =  80  =  320  =  8000  =  63360 

Scale — ascending,  7.92,  25,  4,  80  ;  descending,  80,  4,  25,  7.92. 

The  denomination,  rods,  is  Beidom  used  in  chain  measure,  distances  being 
*ken  in  chains  and  links. 


136 


COMPOUND     NUMBERS. 


11.  Square  Measure. 

143.  A  Square  is  a  figure  having  four  equal  sides, 
and  four  equal  angles  or  corners. 

1  yd. =3  ft. 

1  square  yard  is  a  figure  hav- 
ing four  sides  of  1  yard  or  3  feet 
each,  as  shown  in  the  diagram. 
Its  contents  are  3x3  =  9  square 

feet. 


4^ 

« 

eo 

CO 

II 

II 

"O 

-ri 

>> 

T-t 

T-l 

1yd. =3  ft. 


Thus,  a  square  foot  is  12  inches 
long,  and  12  inches  wide,  and  the 
contents  are  12x12=144  square  inches.  A  surface  20 
feet  long  and  10  feet  wide,  is  a  rectangle,  containing 
20  X  10=200  square  feet.  ^ 

Tlie  contents  or  area  of  a  square,  or  of  any  other  figure 
Jiaving  a  uniform  length  and  a  uniform,  treadth,  is  found, 
ly  multi2)lying  the  length  hy  the  Ireadth. 

144.  Square  Measure  is  used  in  computing  areas  or 
surfaces ;  as  of  land,  boards,  painting,  plastering,  pav- 
ing, etc. 

Table. 

144    Square  Inches  (sq.  in.)  make  1  Square  Foot sq.  ft. 

9    Square  Feet  '*       1  Square  Yard sq.  yd. 


30^  Square  Yards 
160    Square  Rods 
640   Acres 


1  Square  Rod sq.  rd. 

1  Acre A. 

1  Square  Mile sq.  mi. 


A. 

sq.  mi.       1 
1  =  640 


Unit  Equivalents. 

sq.  ft.  sq.  in. 

sq.  yd.                  1  =  144 

sq.  rd.                 1  =                9  =  1376 

1  =         30J^  =         2721  =  39204 

160  =       4840  =       43560  =  6272640 

102400  =  3097600  =  27878400  -  40144896000 


MEASURES     OF     EXTENSIOIT.  137 

Artificers  estimate  their  work  as  follows  : 

By  the  square  foot  :  glazing  and  stone-cutting. 

By  the  square  foot,  or  the  square  yard  :  painting,  plas- 
tering, paving,  ceiling. 

By  the  square  of  100  feet :  flooring,  partitioning,  roof- 
ing, slating,  and  tiling. 

Brick-laying  is  estimated  by  the  thousand  bricks  ;  also 
by  the  square  yard,  and  the  square  of  100  feet. 

1.  In  estimating  the  painting  of  moldings,  cornices,  etc.,  the  measuring  line 
is  carried  into  all  the  moldings  and  cornices. 

2.  In  estimating  brick-laying  by  the  square  yard  or  the  square  of  100  feet,  the 
work  is  understood  to  be  1 14  bricks,  or  12  nches,  thick. 

SuRVETOKS'  Square  Measure. 

145.  This  measure  is  used  by  surveyors  in  computing 
the  area  or  contents  of  land. 

Table. 

625  Square  Links  (sq.  1.)  make  1  Pole P. 

16  Poles  "      1  Square  Chain sq.  ch. 

10  Square  Chains  "      1  Acre A. 

640  Acres  "      1  Square  Mile sq.  mi. 

36  Square  Miles  (6  miles  square)    "      1  Townsliip Tp. 

Unit  Equivalents. 

P.  sq.  1. 

sq.  ch.        1  =      625 
A.      1  =     16  =     10000 
sq.  mi.    1  =    10  =    160  =    100000   ' 
Tp.    1  =   640  =   6400  =    102500  =   64000000 
1  =  36  =  23040  =  230400  =  3686400  =  2304000000 
Scale— ascending,  625,  16,  10,  640,  36  ;  descending,  36,  640,  10, 
16,  625. 

1.  A  equare  mile  of  land  is  also  called  a  section.  — 

2.  Canal  and  railroad  engineers  commonly  use  an  engineer's  chain,  or  a  meas- 
'  uring  tape,  100  feet  long. 


r 


138 


COMPOUIS^D     l^UMBERS. 


III.   Cubic  Measuke. 

146.  A  Cube  is  a  solid,  or  body,  haying   six  equal 

square  sides  or  faces. 

If  each  side  of  a  cube  be  1  yard, 
or  3  feet,  1  foot  in  thickness  of 
this  cube  will  con  tain  3x3x1  =  9 
cubic  feet;  and  the  whole  cube 
will  contain  3x3x3=27  cubic 
^^_^      feet. 

3ft.=iyd.  ^  solid,  or  body,  may  have  the 

three  dimensions  all  alike,  or  all  different.  A  body  4  ft. 
long,  3  ft.  wide,  and  2  ft.  thick  contains  4x3x2=24 
cubic  or  solid  feet.     Hence, 

..  The  cubmor  solid  contents  of  a  'body  are  found  ty  mul- 
'iiplying  the  length,  breadth,  and  thickness  together. 

147.  Cubic  Measure,  also  called  Solid  Measure,  is 
used  in  estimating  the  contents  of  solids,  or  bodies ;  as 
timber,  wood,  stone,  etc. 

Table. 


1728  Cubic  Indies  (cu.  in.) 
27  Cubic  Feet 

40  Cubic  Feet  of  round  timber,  or 
50       "         "      "   hewn 
16  Cubic  Feet 
8  Cord  Feet,  or  ) 
128  Cubic  Feet      f 


24|  Cubic  Feet 


make  1  Cubic  Foot  . . .  .cu.  ft. 
1  Cubic  Yard. . .  .cu.  yd. 

1  Ton  or  Load T. 

1  Cord  Foot cd.  ft. 

1  Cord  of  Wood....Cd. 


[" 


<i 


(( 


i  Perch    of 
1  <.  Stone,    or 
( Masonry. 


.Pch, 


Scale— ascending,  1728,  27,  40,  50,  16,  8, 128,  24| ;    descending, 
24f,  128,  8,  16,  50,  40,  27,  1728. 

1.  A  cubic  yard  of  earth  is  called  a  load. 

2.  Kailroad  and  transportation  companies  estimate  light  freight  by  the  space 
it  occupies  in  cubic  feet,  and  heavy  freight  by  weight. 


MEASURES     OF     CAPACITY.  139 

3.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high,  contains  1  cord;  and 
a  cord  foot  is  one  foot  in  length  of  such  a  pile. 

4.  A  perch  of  stone,  or  of  masonry,  is  W)^  feet  long,  1^  feet  wide,  and  1  foot 
high. 

5.  Embankments  and  excavations  are  estimated  by  the  cubic  yard. 


MEASURES    OF    CAPACITY. 

148.  Capacity  signifies  extent  of  room  or  space. 

All  measures  of  capacity  are  cubic  measures,  solidity 
and  cai^acity  being  referred  to  different  units,  as  will  be 
seen  by  comparing  the  tables. 

Measures  of  capacity  may  be  properly  subdi^-ided  into 
two  classes  :  Measures  of  Liquids,  and  Measures  of  Dry 
Substances. 

I.  Liquid  Measuke.  " 

149.  Liquid  Measure,  also  called  Wine  Measure,  is 
used  in  measuring  liquids  ;  as  liquors,  molasses,  water,  etc. 

^  Table. 

4    Gills  (gi.)  make  1  Pint pt. 

2    Pints  "       1  Quart qt. 

4    Quarts  **       1  Gallon gal. 

31i  Gallons  "       1  Barrel bbl. 

2    Barrels,  or  63  gal.    "       1  Hogshead lilid. 

Unit  Equivalents, 

pt.  gi. 

qt.         1=4 

gal.  1=2=       8 

bbl.      1    =      4  =      8  =      32 

hhd.     1  =  31|-  =  126  =  252  =  1008 

1  =  2  =  63    =  252  =  504  =  2016 

Scale— ascending,  4,  2, 4,  31^,  2 ;  descending,  2,  31^,  4,  2, 4 


r 


140  COMPOUND     NUMBERS. 

150.  The  following  denominations  are  also  in  use  : 

36  Gallons  make  1  Barrel         of  beer. 
54      "        or  li  Barrels  "       1  Hogshead    "      ** 

42      "  "1  Tierce.     . 

3  Hogsheads,  "      1  Pipe  or  Butt. 

2  Pipes,  or  4  Hogsheads,  "      1  Tun. 

1.  The  deDominatioiis,  barrel  and  hogsliead,  are  used  in  estimating  the  ca- 
pacity of  cisterns,  reservoirs,  vats,  etc. 

2.  The  tierce,  hogshead,  pipe,  butt,  and  tun,  are  the  names  of  casks,  and  do  not 
express  any  fixed  or  definite  measures.  They  are  usually  gauged,  and  have  their 
capacities  in  gallons  marked  on  them. 

3.  Ale  or  beer  measure,  formerly  used  in  measuring  beer,  ale,  and  milk,  is 
almost  entirely  discarded. 

4.  The  standard  liquid  gallon  contains  231  cu.  in.,  equal  to  about  8X  11). 
Avoir,  of  pure  water. 


II.  Dry  Measure. 

151.  Dry  Measure  is  used  in  measuring  articles  not 
liquid  ;  as  grain,  fruit,  salt,  roots,  ashes,  etc. 

Table. 

2  Pints  (pt.)  make  1  Quart qt. 

8  Quarts  "      1  Peck , pk. 

4  Pecks  "      1  Bushel.... bu. 

Unit  Equivalents. 

qt.      pt. 

pk.      1  =    2 

bu.     1  =    8  =  16 

1  =  4  =  32  =  64 

Scale — ascending,  2,  8,  4  ;  descending,  4,  8,  2. 

1.  In  England,  8  bu.  of  70  lbs.  each  are  called  a  quarter,  used  in  measuring; 
grain.    The  weight  of  the  English  quarter  is  I4  of  a  long  ton. 

2.  The  wine  and  dry  measures  of  the  same  denomination  are  of  diflferent  ca- 
pacities. The  exact  and  the  relative  size  of  each  may  be  readily  seen  by  the  fol- 
lowing comparative  table : 


TIME.  141 

1.52 •    COMPABATIYE  TABLE  OF  MeASUEES  OP  CAPACITY. 

Cu.  in.  iu      Cu.  in.  in      Cu.  in.  in      Cu.  in.  in 
one  gallon,     one  quart.      one  pint        one  gill. 

Wine  Measure,  231  57|  28|  1-^\ 

Dry  Measure,  (ipk.,)268|-         67^  33|  8-| 

3.  The  beer  gallon  of  282.  inches  is  retained  in  use  only  by  custom.    A  bushe 
is  commonly  estimated  at  2150.4  cubic  inches. 

4.  The  half-peck  or  dry  gallon,  contains  268.8  cubic  inches. 

5.  Six  dry  gallons-  are  equal  to  seven  liquid  gallons. 

6.  A  cubic  foot  of  pure  water  weighs  1000  oz.,  and  equals  62 1  lb.  Avoir. 


Measuee  of  Time. 
153.  Time  is  the  measure  of  duration. 

Table. 

60  Seconds  (sec.)  make  1  Minute min. 

60  Minutes  "  1  Hour h. 

34  Hours  "  1  Day da. 

7  Days  *'  1  Week wk. 

365  Days  "  1  Common  Year yr. 

366  Days  "  1  Leap  Year yr. 

13  Calendar  Months"  1  Year yr. 

100  Years  "      1  Century C. 

Unit  Equivalents. 

h. 
da.    1  = 
wk.    1  =  34  = 

1  =    7  =  168  = 
^*_?2*  _  ^  365  =  8760  =  525600  =  31586000 
^66  =  8784  =  527040  =  31622400 
Scam:— ascending,  60,  60,  34,  7 ;  descending,  7,  24,  60,  60. 


mm. 

sec. 

1  = 

60 

60  = 

3600 

1440  = 

86400 

0080  = 

604800 

142 


COMPOUND     NUMBEES. 


154.  The  calendar  year  is  divided  as  follows  : — 

Season.     No.  of  Month.    Names  of  Months.    Abbreviations.    No.  of  Days. 


Winter 


-    1 


Spring, 


Summer 


Autumn 


Winter, 


■I 


1 
3 

3 

4 
5 

6 

7 
8 

9 

10 
11 

12 


January, 
February, 

Jan. 
Feb. 

31 

28  or  29 

March, 

April, 

May, 

Mar. 
Apr. 
May 

31 
30 
31 

June, 
July, 
August, 

Jun. 
July 

Aug. 

30 
31 
81 

September, 

October, 

November, 

Sept. 

Oct. 

Nov. 

30 
31 
80 

December, 

Dec. 

31 

365  or  366 

1.  The  exact  length  of  a  solar  year  is  365  da.  5  h.  48  min.  46  sec. :  but  for  con- 
venience it  is  reckoned  11  min.  14  sec,  more  than  this,  or  365  da.  6  h.  —  365^4  da. 
This  1^  day,  in  four  years  makes  one  day,  which,  every  fourth,  bissextile,  or  leap 
year,  is  added  to  the  shortest  month,  giving  it  29  days.  The  leap  years  are 
exactly  divisable  by  4,  as  1856, 1860, 1864. 

The  number  of  days  in  each  calendar  month  may  be  easily  remembered  by 
committing  to  memory  the  foUovying  lines : 

"  Thirty  days  hath  September, 
April,  June,  and  November ;  -— 

All  the  rest  have  thirty-one, 
Save  February,  vrhich  alone 
Hath  twenty-eight ;  and  one  day  more 
We  add  to  it  one  year  in  four." 

2.  In  most  business  transactions  30  days  are  called  1  month. 

3.  The  centuries  are  numbered  from  the  commencement  of  the  Christian  era; 
the  months  from  the  commencement  of  the  year ;  the  days  from  the  commence^ 
ment  of  the  month,  and  the  hours  from  the  commencement  of  the  day  (12  o'clock, 
midnight).  Thus,  May  23d,  1860,  9  o'clock  A.  M.,  is  the  9th  hour  of  the  23d  day 
of  the  5th  month  of  the  60th  year  of  the  19th  century. 


circular   measure.  143 

Circular  Measure. 

155.  Circular  Measure,  or  Circular  Motion,  is  used 
principally  in  surveying,  navigation,  astronomy,  and 
geography,  for  reckoning  latitude  and  longitude,  deter- 
mining locations  of  places  and  vessels,  and  computing 
difference  of  time. 

Each  circle,  great  or  small,  is  divisible  into  the  same 
number  of  equal  parts,  as  quarters,  called  quadrants^ 
twelfths,  called  signs,  360ths,  called  degrees,  etc.  Con- 
sequently the  parts  of  unequal  circles,  although  having 
the  same  names,  are  of  unequal  lengths. 

Table. 

60  Seconds  (")        make  1  Minute ^ 

60  Minutes  "      1  Degree °. 

30  Degrees  "      1  Sign S. 

12  Signs,  or  360'       "      1  Circle C. 


\ 


Unit  Eqtjivalents. 

'  // 

°  1  =  60 

S.         1  =       60  =       3600 

C.   1  =  30  =  1800  =  108000 

1  =  12  =  360  =  21600  =  1296000 

Scale— ascending,  60,  60,  30,  12  ;  descending,  12,  30,  60,  60. 

1.  Minutes,  of  the  earth's  circumference,  are  called  geographic  or  nautical 
miles. 

2.  The  denomination,  signs,  is  confined  excki?ively  to  Astronomy. 

3.  A  degree  has  no  fixed  linear  extent.  When  applied  to  any  circle,  it  is 
always  -^^-^  part  of  the  circumference.  But,  strictly  speaking,  it  is  not  any  part 
of  a  circle. 

4.  90°  make  a  quadrant  or  right-angle. 
6.  60°  make  a  sextant  or  \  of  a  circle. 


144  COMPOUND     NUMBEES. 

MISCELLANEOUS    TABLES, 

156.    C0Uli[TIKG. 


• 


12  Units  or  things  make  1  Dozen doz. 

12  Dozen  "      1  Gross gro. 

12  Gross                      *'      1  Great  gross. . .  G.  gro, 
20  Units  ♦•      1  Score sc. 

157.  Papee. 

24  Sheets make 1  Quire qre. 

20  Quires  "  1  Ream rm. 

2  Reams  "  1  Bundle bdl. 

5  Bundles  "  1  Bale B. 

158.  Books. 

The  terms  folio,  quarto,  octavo,  duodecimo,  etc.,  indicate 
the  number  of  leayes  into  which  a  sheet  of  paper  is  folded. 

When  a  sheet  is  The  book  is  And  1  sheet  of 

folded  into  called  paper  makes 

2  Leaves  a  Folio,  4  pp.  (pages). 

4      "  a  Quarto  or  4to,  8    " 

8      '*  an  Octavo  or  8 vo,  16    " 

12      '*  a  Duodecimo  or  12mo,  24     " 

16      "  alGmo,  32    " 

18      "  anl8mo,  86 


it 


159.  CoPYiisrG. 

75  Words  make  1  Folio  or  sheet  of  Common  Law, 
190      "  **      1     "      "      "       "   Chancery. 

160.  An  Aliquot  Part  of  a  number  is  such  a  part  as 
will  exactly  divide  that  number;  thus,  3,  5,  7|-  are  aliquot 
parts  of  15. 

An  aliquot  part  may  be  a  whole  or  mixed  number,  while  a  factor  must  be  a 
whole  number. 


ALIQUOT     PAETS. 


145 


161.        Aliquot  Paets  of  Oi^e  Dollae. 


50  cents 
334-  cents 
25  cents 
20  cents 
16-|  cents 


hot 
iof 


dollar, 
dollar, 
dollar, 
dollar, 
dollar. 


12-J-  cents  = 

10    cents  = 

8^  cents  = 

6^  cents  = 

5    cents  = 


of 
of 


1 

"§■ 
1 

1  0 

-hot 
yVof 
2V  of 


dollar, 
dollar, 
dollar, 
dollar, 
dollar. 


163. 

160  Eods 
80  Eods 
40  Eods 


Aliquot  Paets  of  a  Mile. 


—    8 


i  Mile. 
i  Mile. 
4  Mile. 


1760  Feet 
880  Feet 
440  Feet 


=  i  Mile. 
=  i  Mile. 


163.  Aliquot  Paets  of  an  Acee. 


80  Square  Eods  =  -J-  Acre. 
40  Square  Eods  =  ^  Acre. 


32  Square  Eods  =  ^  Acre. 
20  Square  Eods  : 


-J-  Acre. 


164. 


Aliquot  Paets  of  a  Ton". 


10  Hund.  lbs.  =  ^  Ton. 
5  Hund.  lbs.  =  i  Ton. 
4  Hund.  lbs.  =  ^  Ton. 


2  Hund.  50  lb. 
2  Hund.  lbs. 
1  Hund.  lbs. 


i  Ton. 
Ton. 


1 0 

A  Ton. 


165.  Aliquot  Paets  of  a  Poukd  Atoiedupois. 


8  Ounces 


^  Pound. 


4  Ounces  =  ^  Pound. 


2  Ounces  = 
1  Ounce    = 


JL 

8 

1 

16 


Pound. 
Pound. 


166.  Aliquot  Paets 

Parts  of  One  Year. 

6    Months  =  ^  Year. 
4   Months  = 
3    Months  == 
2    Months  = 
1-J  Months  = 


—     6 


-J-  Year. 
^  Year. 
4-  Year. 


■i  Year. 


8 


1    Month    =  -^  Year. 
10 


OF  Time. 

Parts  of 

15  Days 
10  Days 

6  Days 

5  Days 

3  Days  =  ^  Month. 

2  Days 


'  One  Month. 

J  Month. 
I  Month. 
I    Month. 


—       6 


i   Month. 


^  Month. 


IT 


146  COMPOUND     K  UMBERS. 

REDUGTIOISr. 

167.  Reduction  is  the  process  of  changing  the  de^ 
nomination  of  a  number,  without  altering  its  value. 

168.  Reduction  Descending  is  changing  a  number 
of  one  denomination  to  another  denomination  of  less  unit 
value,  and  is  performed  by  multiplication;  thus,  $1  =  10 
dimesrrlOO  cents=1000  mills  ;  lyd.  =  3  feet=36  in. 

I.  Reduce  6  gal.  2  qt.  1  pt.  to  pints. 

OPERATION.  Analysis.    Since  in  1  gal.  there 

6  gal.  2  qt.  1  pt.    •    are  4  qt.,  in  6  gal.  there  are  4  qt.  x  6 
^  =24  qt.,  and  the  2  qt.  in  the  given 

—  number  added,  makes  26  qt.  in  6 

'^^  ^'  gal.  2  qt.    Since  in  1  qt.  there  are  2 

2  qt.,  in26qt.  there  are2pt.  x26=52 

Ans    53  T)t.  pt.,and  the  1  pt.  in  the  given  num- 

ber added,  make  53  pints  in  the 
given  compound  number.  As  either  factor  may  be  used  as  a  mul- 
tiplicand (61),  we  may  consider  the  numbers  in  the  descending 
scale  as  multipliers. 

Rule.  I.  Multiply  the  highest  denomination  of  the  given 
numher  hy  that  iiumher  of  the  scale  which  will  reduce  it 
to  the  next  loiver  denomination,  and  add  to  the  product  the 
given  number,  if  any,  of  that  lower  denomination, 

II.  Proceed  in  the  same  manner  with  the  result  obtained 
171  each  lower  denomination,  until  the  reduction  is  brought 
to  the  denomination  required. 

Examples  foe  Peactice. 

2.  In  8  lb.  10  oz.  how  many  ounces  ?        Ans.  138  oz. 

3.  In  £12  6s.  9d.  how  many  pence  ?  Ans.  2961d. 

4.  In  4  yd.  1  ft.  10  in.  how  many  inches  ? 

5.  In  3  mi.  226  rd.  how  many  rods  ? 


EEDUCTIOiq".  147 

6.  In  18s.  8d.  3  far.  how  many  farthings  ? 

Ans.  899  far. 

7.  Reduce  3  lb.  9  oz.  12  pwt.  to  pennyweights. 

8.  In  2  hhd.  15  gal.  2  qt.  how  many  pints  ? 

9.  Reduce  4  da.  5  hr.  to  minutes.      A}is,  6060  min. 

10.  Reduce  10  bu.  1  pk.  6  qt.  to  pints.     Ans.  668  pt, 

11.  Reduce  14  A.  140  sq.  rd.  to  square  rods. 

12.  Reduce  4  cd.  3  cd.  ft.  9  cu.  ft.  to  cubic  inches. 

13.  Reduce  4  yr.  7  mo.  to  hours.  A7is.  39600  hr. 

14.  Change  2  T.  11  cwt.  to  pounds.       Ans.  5100  lb. 

15.  Change  9  lb.  9  oz.  10  pwt.  to  grains. 

16.  Change  5  lb.  6  §  4  3  2  3  10  gr.  to  grains. 

17.  Change  3  mi.  240  rd.  to  feet.  Ans.  19800  ft. 

18.  In  40  chains  how  many  links  ?  Ans.  4000  1. 
^19.  In  28  sq.  rd.  12  sq.  yd.  4  sq.  ft.  how  many  square 

inches?  Ans.  1113840  sq.  in. 

20.  In  16  A.  4  sq.  ch.  8  P.  80  sq.  1.  how  many  square 
Hnks  ?  Ans.  1645080  sq.  1. 

21.  In  12  tons  of  round  timber  how  many  cubic  inches? 

22.  In  8  bbl.  26  gal.  how  many  pints  ?  Ans.  2224  pt. 

23.  Reduce  4  pipes  to  quarts.  A7is.  2016  qt. 

24.  Reduce  23  bu.  3  pk.  to  pints.  A^is.  1520  pt.  ' 

25.  Reduce  8  S.  18°  40'  to  minutes.         Ans.  15520'. 

26.  Reduce  15°  to  seconds.  Ans.  54000". 

27.  Reduce  2  months  to  minutes.     Ans.  86400  min. 

28.  Change  2  reams  10  quires  to  sheets. 

29.  In  40  score  how  many  single  things  ?    Ans.  800. 

30.  In  14  great  gross  how  many  dozens  ? 

31.  In  30°  20'  24"  how  many  seconds  ? 

32.  In  the  3  Autumn  months  how  many  hours  ? 

33.  In  the  three  Summer  months  how  many  minutes  ? 

34.  In  75  cords  how  many  cubic  feet  ? 


148  COMPOUND     NUMBERS. 

169.  Reduction  Ascending  is  changing  a  number  of 
one  denomination  to  another  of  greater  unit  value,  and  is 
performed  by  Division;  thus,  1000  mills=100  cents=$l. 

1.  Kednce  53  pints  to  gallons. 

OPEEATiOJT.  Analysis.    Dividing  the  given 

2  )  53  number  of  pints  by  2,  because 

,  r~~    ,        ^      ,  there   are  4   as  many  quarts  as 

4)26qt.  +  lpt.  .   ,  L  •     OP       1     1 

z 1      '       ^  pmts,  we  obtain  26  quarts  plus  a 

6  gal.  -\-2  qt.  remainder  of  1   pt.     Next  divide 

Ans.  6  gal.  2  qt.  1  pt.       ^6  quarts  by  4,  because  there  are 

^  as  many  gallons  as  quarts,  and 
we  obtain  6  gallons  and  a  remainder  of  2  qt.  This  last  quotient, 
with  the  several  remainders  annexed,  forms  the  answer. 

2.  Eeduce  4902  inches  to  rods. 

OPERATION.  Analysis.     Divide  successively 

12  )  4902  ^J  the  numbers  in  the  ascending 

-,  n  i~\~7^  j^i.    ,   n  '  scale  in  the  same  manner  as  in  the 

16-t)  408  it.4-6  m.  ,.  i      -d  *  •     ^• 

i)  preceding   example.     But  in    di- 

-  viding  the  408  ft.  by  16|,  first  re- 

83  )  816  duce  408  ft.  to  halves  by  multiply- 

24  rd  4-J-4 12  ft     ^^^  ^^  ^'  ^^^^  ^^  have  816  halves  ; 

A         c^A      T    -^  c-f  j»j    />  •  ^iid   reducing    16^  to  halves,  we 

Ans.  24  rd.  12  ft.  6  m.     .         oo  ^  ,        "mr        ..  • -,. 

have  33  halves.      Then  dividing 

816  by  33  we  obtain  24  rd.  plus  a  remainder  of  24  halves=to  12  ft., 

which,  with  the  preceding  remainder  annexed  to  the  last  quotient, 

,  gives  the  answer. 

EuLE.  I.  Divide  the  given  nuniber  hy  that  mimler  of 
the  scale  which  will  reduce  it  to  the  next  higher  denomina- 
tion, 

II,  Divide  the  quotient  hy  the  next  higher  nuniber  in  the' 
scale  ;  and  so  proceed  to  the  highest  denomination  required. 
The  last  quotient,  with  the  several  remainders  annexed  in 
a  reversed  order,  will  he  the  answer. 


reduction.  149 

Examples  for  Practice. 

3.  How  many  pounds  in  3460  ounces  ? 

Ans.  216  lb.  4  oz. 

4.  How  many  shillings  in  556  farthings  ? 

Ans.  lis.  7d. 

5.  How  many  yards  in  1242  inches  ? 

6.  How  many  gallons  in  2347  pints  ? 

7.  Reduce  23547  troy  grains  to  pounds. 

A71S.  4  lb.  1  oz.  1  pwt.  3  gr. 

8.  Reduce  1597  quarts  to  bushels. 

Ans.  49  bu.  3  pk.  5  qt. 

9.  Reduce  107520  oz.  avoirdupois  to  pounds. 

10.  In  28635  sec.  how  many  hours  ? 

Ans.  7  hr.  57  min.  15  sec. 

11.  In  10000"  how  many  degrees  ? 

A71S.  2°  46'  40". 

12.  In  11521  gr.  apothecaries  weight  how  many  pounds? 

Ans.  2  ft)  1  gr. 

13.  In  3561829  seconds  how  many  weeks  ? 

14.  Reduce  67893  cu.  ft.  to  cords. 

15.  In  1491  pounds  how  many  hundred  weight  ? 

16.  In  12244  pints  how  many  hogsheads  ? 

17.  In  25600  sq.  rd.  how  many  acres  ?      Ans.  160  A. 

18.  How  many  miles  in  51200  rd.  ?         Ans.  160  mi. 

19.  How  many  barrels  in  6048  gills  ?        Ans.  6  bbl. 

20.  In  316800  inches  how  many  miles  ?     Ans.  5  mi. 

21.  In  1728  how  many  gross  ?  Ans.  12  gross. 

22.  In  4060  how  many  score  ?  A7is.  203  score. 

23.  Reduce  1435  feet  to  fathoms. 

24.  Reduce  10000  sheets  of  paper  to  reams. 

Ans.  20  reams  16  quires  16  sheets. 

25.  Reduce  27878400  sq.  ft.  to  square  miles. 


150  compound  kumbees. 

Promiscuous  Examples  in  Eeduction. 

1.  Eeduce  4  dollars  67  cents  to  cents.    Ans.  467  cents. 

2.  Eeduce  3724  mills  to  dollars.  Ans.  $3,724. 

3.  Eeduce  9690  cents  to  dollars.  Ans.  $96.90. 

4.  Eeduce  8  dollars  to  mills.  Ans.  8000  mills. 

5.  In  91751  farthings  how  many  pounds  ? 

Ans.  £95  lis.  5d.  3  far. 

6.  In  3  lb.  4  oz.  7  pwt.  how  many  grains  ? 

7.  In  3  tons  of  cheese  how  many  pounds  ? 

8.  How  much  will  4  cheese  cost,  each  weighing  36 
pounds,  at  9  cents  a  pound?  Ans.  112.96. 

9.  How  much  would  2  lb.  8  oz.  12  pwt.  of  gold  dust  be 
worth,  at  72  cents  a  pwt.  ?  Ans.  1469.44. 

10.  Bought  1  T.  15  cwt.  36  lb.  of  sugar  at  7  cents  a 
pound  ;  what  did  it  cost  ?  Ans.  $247.52. 

11.  Paid  $25.50  for  one  hogshead  of  molasses,  and  sold 
it  all  at  50  cents  a  gallon  ;  what  was  the  whole  gain  ? 

12.  How  many  pounds  in  6  barrels  of  flour  ? 

13.  How  many  bushels  of  oats  in  a  load  weighing  1280 
pounds  ?  Ans.  40  bu. 

14.  How  many  bushels  of  wheat  in  a  load  weighing 
2175  pounds  ?  Ans.  36  bu.  15  lb. 

15.  A  grocer  bought  3  barrels  of  flour  at  16  a  harrel, 
and  sold  it  out  at  4  cents  a  pound  ;  what  did  he  gain  on 
the  whole  ?  Ans.  $5.52. 

16.  In  a  board  12  feet  long  and  2  feet  wide,  how  many 
square  feet  ?  Ans.  24  sq.  ft. 

17.  In  a  block  of  marble  6  feet  long  and  3  feet  square, 
how  many  cubic  feet  ?  Ans.  54  cu.  feet. 

18.  In  a  pile  of  wood  26  feet  long,  6  feet  high  and  3 
feet  wide,  how  many  cubic  feet  ?  how  many  cords  ? 

A71S.  468  cu.  ft.;  or  3  Cd.  84  cu.  ft. 


REDUCTIOIs^.  151 

19.  In  259200  cubic  inches  of  he^^Ti  timber  bow  many 
tons  ?  Ans.  3  0^. 

20.  How  many  square  rods  in  a  field  90  rods  long  and  75 
rods  wide?   How  many  acres?        A?is.  42  A.  30  sq.  rd. 

21.  A  pond  of  water  measures  3  fathoms  2  feet  9  inches 
in  depth  ;  how  many  inches  deep  is  it  ?      Ans,  249  in. 

22o  What  will  3  miles  of  telegraph  cable  cost  at  12 
cents  a  foot  ?  Ans.  $1900.80. 

23.  What  is  the  age  of  a  man  3  score  and  5  years  old  ? 

Ans.  65  years. 

24.  How  much  will  I  receiye  for  a  load  of  wheat  weigh- 
ing 2760  pounds  at  $1.50  per  bushel  ?  Ans.  869. 

25.  How  many  cubic  feet  in  a  stick  of  timber  32  feet 
long,  2  feet  wide,  and  1  foot  thick  ?        Ans.  64  cu.  ft. 
^26.  How  many  square  feet  in  1  acre  ? 

27.  In  176  yards  how  many  rods?  Ans.  32  rd. 

28.  A  pile  of  wood  is  16  feet  long,  8  feet  high,  and  8 
feet  wide  ;  how  much  is  it  worth  at  $3.50  a  cord  ? 

Ans.  $28. 

29.  What  would  be  the  value  of  a  city  lot  40  feet  wide 
and  120  feet  long,  at  2  cents  a  square  foot  ?    A?is.  $96. 

30.  A  grocer  bought  4  barrels  of  cider,  at  $2  a  barrel, 
and  after  converting  it  into  vinegar,  he  retailed  it  at  15 
cents  a  gallon  ;  what  was  his  whole  gain  ?  Ans.  $10.90. 

31.  At  6  cents  a  pint,  how  much  molasses  can  be  bought 
for  $4.26  ?  Ans.  8  gal.  3  qt.  1  pt. 

32.  An  innkeeper  bought  a  load  of  40  bushels  of  oats, 
at  36  cents  a  bushel,  and  retailed  them  at  25  cents  a  peck; 
what  did  he  make  on  the  load?  Ans.  $25.60. 

33.  What  will  be  the  cost  of  a  hogshead  of  wine  at  8 
cents  a  gill?  Ans.  $161.28. 

34.  In  120  gross  how  many  score  ?      Ans.  864  score. 


152  COMPOUND     l^UMBERS. 

35.  If  a  man  walk  4  miles  an  hour,  and  10  hours  a  day, 
how  many  miles  can  he  walk  in  24  days  ?    Ans.  960  mi. 

36.  What  will  be  the  cost  of  2  hu.  1  pk.  6  qt.  of  timo- 
thy seed,  at  10  cents  a  quart  ?  Ans.  $7.80, 

37.  What  would  be  the  value  of  a  silyer  goblet,  weigh- 
ing 8  oz.  14  pwt.,  at  $.15  a  pwt.  ?  Ans.   126.10. 

38.  What  will  16  reams  of  paper  cost  at  20  cents  a 
quire  ?  Ans.  $64. 

39.  If  1  bushel  of  -^heat  make  45  pounds  of  flour,  how 
many  pounds  will  500  bushels  make  ?   How  many  barrels  ? 

Ans.  114  bbl.  156  pounds. 

40.  Bought  a  gold  chain,  weighing  2  oz.  18  pwt.,  at  1.90 
a  pwt. ;  what  did  it  cost  ?  Ans.  $52.20. 

41.  How  many  minutes  more  are  there  in  the  Summer 
than  in  the  Autumn  months  ?  Ans.  1440  min. 

42.  How  much  will  it  cost  to  dig  a  cellar  24  ft.  long, 
18  ft.  wide,  and  6  feet  deep,  at  1  cent  a  cubic  foot  ? 

Ans.  $25.92. 

43.  How  many  boxes,  each  containing  12  pounds,  can 
be  filled  from  a  hogshead  of  sugar  containing  9  cwt.  ? 

Ans.  75  boxes. 

44.  What  will  be  the  cost  of  5  bales  of  cloth,  each  bale 
containing  15  pieces,  and  each  piece  measuring  26  yards, 
at  $1.75  a  yard  ? 

45.  If  a  cannon  ball  goes  at  the  rate  of  10  miles  a  min- 
ute, how  many  miles  would  it  go,  at  the  same  rate,  in  2 
hours  ?  A71S.  1200  miles. 

46.  At  11  cents  a  pound,  what  will  be  the  cost  of  3  cwt. 
71  lb.  of  sugar?  Ans.  $40.81. 

47.  If  a  man  earn  $30  a  month,  how  much  will  he  earn 
in  5  years?  Ans.  $1800. 


ADDITION.  153 


ADDITION. 

170.  Compound  numbers  are  added,  subtracted,  multi- 
plied, and  divided  by  the  same  general  method  as  are  em- 
ployed in  simple  numbers.  The  only  modification  of  the 
operations  and  rules  is  that  required  for  borrowing,  carry- 
ing, and  reducing  by  a  varying,  instead  of  a  uniform  scale. 

1.  What  is  the  sum  of  36  bu.  2  ^jk.  6  qt.  1  pi,  25  bu. 
1  pk.  4  qt.,  18  bu.  3  pk.  7  qt.  1  pt.,  9  bu.  0  pk.  2  qt.  1  pt.  ? 

Analysis.  Arranging  the 
numbers  in  columns,  jDlacing 
units  of  tlie  same  denomina- 
tion under  each  other,  we  first 
add  the  units  in  the  right-hand 
column,  or  lowest  denomina- 
Ans.    90  0  4  1  tion,  and  find  the  amount  to 

be  3  pints,  which  is  equal  to 
1  qt.  1  pt.  Write  the  1  pt.  under  the  column  of  pints,  and  add  the 
1  qt.  to  the  column  of  quarts.  We  find  the  amount  of  the  second 
column  to  be  20  qt.,  which  is  equal  to  2  pk.  4  qt.  Writing  the  4  qt. 
under  the  column  of  quarts,  add  the  2  pk.  to  the  column  of  pecks. 
Adding  the  column  of  pecks  in  the  same  manner,  we  find  the 
amount  to  be  8  pk.,  equal  to  2  bu.  Writing  0  pk.  under  the  column 
of  pecks,  add  the  2  bu.  to  the  column  of  bushels.  Adding  the  last 
column,  we  find  the  amount  to  be  90  bu.  which  we  write  under  the 
left-hand  denomination,  as  in  simple  numbers. 

EuLE.  I.  Write  the  numbers  so  that  those  of  the  same 
unit  value  stand  in  the  same  column. 

II.  Beginning  at  the  right  hand,  add  each  denomination 
as  in  simple  numbers,  carrying  to  each  succeeding  denomi- 
nation one  for  as  many  units  as  it  talces  of  the  denomina- 
tion added,  to  make  one  of  the  next  higher  denomination. 


OPERATION. 

bu. 

pk. 

qt. 

pt. 

36 

2 

6 

1 

25 

1 

4 

0 

18 

3 

7 

1 

9 

0. 

2 

1 

154 


COMPOUKD     NUMBEES. 


Examples  for  Practice. 


£. 

(2.) 
s.   d. 

far. 

47 

10   9 

1 

25 

6   4 

3 

36 

18   0 

2 

12 

0  10 

0 

8 

7   3 

1 

130 

3   3 

w 
3 

lilid. 
24 

(4.) 
gal.  qt. 

21    3 

pt. 
1 

102 

42   2 

0 

38 

9   0 

1 

42 

50   1 

0 

i 

207 

60   3 

0 

da. 

(6.) 
h.  min. 

sec. 

27 

14  40 

36 

106 

20  14 

25 

16 

12  50 

45 

52 

16  39 

18 

mi. 

(8.) 
rd.   yd. 

ft. 

in. 

2 

25   4 

1 

10 

1 

30   1 

2 

7 

4 

16   6 

0 

4 

10 

8-   2 

2 

11 

(3.) 


R.  § .    3 • 

3.  gr. 

10  10   4 

1  12 

9   5 

2  10 

14   4   0 

0  16 

6   0   7 

1   0 

6   3 

2  15 

32   7   5 

2  13 

(5.) 

T.  cwt.  lb. 

oz. 

3  12  15 

10 

16  20 

7 

5   9   6 

0 

18  17 

14 

10  15  59 

15 

(7.) 

lb.  oz.  pwt. 

gr. 

16  11  18 

21 

26   9  15 

10 

11  10   0 

8 

4   6  12 

0 

(9.) 

P.  sq.  yd. 

sq.  ft. 

12   20 

5 

9   15 

6 

15   10 

7 

20   26 

3 

ADDITIO]S".  155 

10.  Wliat  is  the  Slim  of  2S.  12°  40'  25",  5S.  9°  27'  38", 
16°  10' 50",  IS.  16°? 

11.  What  is  the  sum  of  44A.  104P.,  lOA.  20P.,  25A. 
40P.,  6A.  36P.?  Ans.  86A.  40P. 

12.  What  is  the  sum  of  25  rd.  12  ft.  5  in.,  28  rd.  9  ft. 

10  in.,  18  rd.  10  ft.,  12  rd.  14  ft.  9  in.  ? 

A71S.  85  rd.  14  ft. 

13.  What  is  the  sum  of  5  Cd.  6  cd.  ft.  9  cu.  ft.,  4  Cd.  3 
cd.  ft.  12  cu.  ft.,  10  Cd..l4  cu.  ft.,  2  Cd.  7  cd.  ft.  ? 

A71S.  23  Cd.  2  cd.  ft.  3  cu.  ft. 

14.  Wliat  is  the  sum  of  40  yd.  2  ft.  10  in.,  37  yd.  1  ft.  9 
in.,  28  yd.  11  in.,  10  yd.  2.  ft.,  15  yd.  ? 

A71S.  132  yd.  1  ft.  6  in. 

15.  What  is  the  sum  of  13  Cd.  60  cu.  ft.  164  cu.  in.,  25 
Cd.  75  cu.  ft.,  18  Cd.  25  cu.  ft.  540  cu.  in.,  8  Cd.  1030  cu. 
in.  ?  Ans.  65  Cd.  33  cu.  ft.  6  cu.  in. 

16.  A  grocer  bought  4  hhd.  of  sugar  ;  the  first  weighed 

11  cwt.  71  lb.  ;  the  second  10  cwt.  41  lb. ;  the  third  10 
cwt.  22  lb. ;  and  the  fourth  9  cwt.  75  lb.  How  much  did 
the  whole  weigh  ?  Ans.  2T.  2  cwt.  9  lb. 

17.  A  man  has  a  farm  divided  into  three  fields ;  the  first 
contains  26A.  HOP.  ;  the  second,  48A.  27P. ;  and  the 
third,  35A.  80P.     How  many  acres  in  the  farm  ? 

Ans.  IIOA.  57P. 

18.  If  a  printer  one  day  use  2  bundles  1  ream  10  quires 
of  paper,  the  next  day  3  bundles  1  ream  12  quires  20 
sheets,  and  the  next,  4  bundles  9  quires,  hovr  much  does 
he  use  in  the  three  days  ? 

A71S.  10  bundles  1  ream  11  quires  20  sheets. 

19.  A  tailor  used,  in  one  year,  3  gross  6  doz.  10  buttons, 
another  year,  2  gross  9  doz.  9  buttons,  and  another  year, 
4  gross  7  doz. ;  how  m-any  did  he  use  in  the  three  years  ? 


OPERATION. 

lb. 

OZ. 

pwt. 

gr. 

24 

6 

5 

12 

14 

9 

10 

7 

156  COMPOUND     KUMBEKS. 


SUBTKACTION. 

171.  From  24  lb.  6  oz.  5  pwt.  12  gr.  take  14  lb.  9  oz. 
10  pwt.  7  gr. 

Analysis.  Writing  the 
Bubtraliend  under  the  min- 
uend, placing  units  of  the 
same  denomination  under 

Ans,  "9        8        15  5         f '^  ""f^"'  ^^^i"^'*  ^  ^'- 

from  12  gr.  and  write  the 

remainder,  5  gr.,  underneath.  Since  we  cannot  subtract  10  pwt- 
from  2  pwt.,  add  1  oz.  or  20  pwt.  to  the  5  pwt.,  and  subtract  10  pwt. 
from  the  sum,  25  pwt.,  and  write  the  remainder,  15  pwt.,  under- 
neath. Having  added  20  pwt.  or  1  oz.  to  the  minuend,  we  now  add 
1  oz.  to  the  9  oz.  in  the  subtrahend,  making  10  oz, ;  but  as  we  can- 
not take  10  oz.  from  6  oz  we  add  1  lb.  or  12  oz.  to  the  6  oz. ,  making 
18  oz.,  and  subtracting  10  oz.  from  18  oz.  we  write  the  remainder,  8 
oz.,  under  the  denomination  of  ounces.  Having  added  1  lb.  to  the 
minuend,  we  now  add  1  lb.  to  the  14  lb.  in  the  subtrahend,  and 
subtracting  15  lb.  from  24  lb.  as  in  simple  numbers,  we  write  the 
remainder,  9  lb.,  under  the  denomination  of  pounds. 

EuLE.  I.  Write  the  subtrahend  under  the  minuend, 
so  that  units  of  the  same  denomination  stand  under  each 
other. 

II.  Beginning  at  the  right  hand,  suMract  each  denomi- 
nation separately,  as  in  simple  numbers. 

III.  If  the  number  of  any  denomination  in  the  subtra- 
hend exceed  that  of  the  same  denomination  in  the  minuend, 
add  to  the  nwnber  in  the  minuend  as  many  units  as  make 
one  of  the  next  higher  denomination,  and  then  suhtract ;  in 
this  case  add  1  to  the  next  higher  denomination  of  the  sub- 
trahend before  subtracting.  Proceed  in  the  same  manner 
with  each  denomination. 


SUBTRACTIOiq'.  157 


Examples 

FOR 

Practice. 

1 

From 

(2.) 
cwt.   qr. 
18   1 

lb. 

14 

oz. 
9 

(3.) 

hbd.  gal.   qt.  pt. 

7   28   2   1 

Take 

5   2 

20 

6 

3   42   3   0 

Rem. 

12   2 

22 

3 

3   48   3   1 

ft. 
12 

1.  3. 
7   3 

3. 
1 

gr. 
11 

(5.) 
bu.   pk.  pt.  pt 
104   2   6   0 

8 

5   4 

2 

15 

56   3   4   1 

ml. 

(6.) 
rd.   yd.   ft. 

in. 

(7.) 
A.    P. 

40 

130   3 

2 

10 

400   125 

14 

115   4 

1 

1 

325   130 

wk. 

(8.) 
da.   hr. 

min. 

sec 

(9.) 

S.   °    '    " 

10 

4   16 

40 

22 

6   25   45   38 

4 

5   12 

45 

50 

4   28   40   50 

T. 

(10.) 
cwt.   lb. 

oz. 

(11.) 
Cd.  cd.  ft.  cu.ft.  cu.  in. 

14 

5   68 

9 

120   4    6   520 

10 

14   82 

14 

94   7   12  1500 

yd. 

74 

(12.) 
ft.  in. 
2   6 

0^ 
Cd. 

325 

5.) 
cu.  ft. 

80 

(14.) 

sq.  yd  sq.  ft.  sq.  in. 

27    6    96 

9 

2   9 

128 

112 

14    8   120 

158  COMPOUND     KUMBERS. 

15.  From  135  mi.  240  rd.  take  90  mi.  185  rd. 

Ans.   35  mi.  55  rd. 

16.  A  man  bonglit  1  lihd.  of  molasses,  and  sold  42  gal. 
3  qt.  1  pt. ;  how  much  remained?        Ans.  20  gal.  1  pt. 

17.  A  person  bought  9  T.  14  cwt.  3  qr.  of  coal,  and 
having  burned  4  T.  15  cwt.  sold  the  remainder";  how 
much  did  he  sell  ?  Ans.  4=  T.  19  cwt.  3  qr. 

18.  If  from  a  tub  of  butter  containing  1  cwt.  21  lb. 
there  be  sold  24  lb.  8  oz.  how  much  remains  ? 

Ans.  96  lb.  8  oz. 

19.  From  a  pile  of  wood  containing  42  Cd.  5  cd.  ft.  there 
was  sold  16  Cd.  6  cd,  ft.  12  cu.  ft. ;  how  much  remained  ? 

A71S.  25  Cd.  6  cd  ft.  4  cu.  ft. 

20.  If  from  a  field  containing  37  A.  146  P.  there  be 
taken  14  A.  110  P.,  how  much  will  there  be  left? 

21.  A  farmer  having  raised  50  bu.  2  pk.  of  wheat,  kept 
for  his  own  use  25  bu.  3  jjk.  ;  how  much  did  he  sell  ? 

A71S.  24  bu.  3  pk. 

22.  The  distance  from  'New  York  to  Albany  is  150 
miles ;  when  a  man  has  traveled  84  mi.  270  rd.  of  the 
distance,  how  much  farther  has  he  to  travel  ? 

Ans.  65  mi.  50  rd. 

23.  "What  is  the  difference  in  the  longitude  of  two 
places,  one  71°  20'  26",  and  the  other  44°  35'  58"  West? 

Ans.  26°  44'  28". 

24.  If  from  a  hogshead  of  molasses  10  gal.  2  qt.  be 
drawn  at  one  time,  9  gal.  3  qt.  at  another,  and  14  gal.  at 
another,  how  much  will  remain  ?        A?i s.  2S  gal.  3  qt. 

25.  From  a  section  of  land  containing  640  acres,  there 
was  sold  at  one  time  140  A.  116  P.,  at  another  time  200 
A.  40  P.,  and  at  another  time  75  A.  28  P.  ;  how  much 
remained  ?  Aiis.  223  A.  136  P. 


MULTIPLICATI02S'.  159 


MULTIPLICATION, 

173.    1.  A  farmer  has  8  fields,  each  containing  4  A 
107  P. ;  how  much  land  in  all  ? 

OPERATION.  Analysis.     In  8  fields  are  8  times  as 

A.        P.  much  land  as  in  1  field.     We  write  the 

4       107  multiplier  under  the  lowest  denomination 

8  of  the  multiplicand,  and  proceed  thus :  8 

"^^Z  T^  times  107  P.  are  85G  P.,  equal  to  5  A.  56  P.  ; 

and  we  write  the  56  P.  under  the  number 
multiplied.  Then,  8  times  4  A.  are  32  A.,  and  5  A.  added  make  37 
A.,  which  we  write  under  the  same  denomination  in  the  multipli- 
cand, and  the  work  is  done. 

KuLE.     I.    Write  tlie  multiplier  under  the  lotvest  de7iom- 
ination  of  the  multiplicand. 

II.  Multiply  as  in  siinple  numbers,  and  carry  as  in  ad' 
dition  of  compound  numbers. 

Examples  for  Practice. 

(2.) 
hhd.     gal.     qt.     pt. 

6        20        2       1 
3 


Ans.  18       61 


(4.) 
lb.    oz.    pwt.    gr. 

12     8       14     16 

5^ 

63    7      13      8 


bu. 
9 

(3.) 
pk.     qt. 

2        6 

pt. 

1 

4 

38 

3       2 

0 

T. 

10 

(5.) 

cwt.     lb. 

15     20 

oz. 

8 

6 

64 

14     23 

0 

160  COMPOUND     KUMBEES. 

6.  Multiply  14  A.  106  P.  by  8.      Ans.  117  A.  48  P. 

7.  Multiply  6  yd.  2  ft.  9  in.  by  12.  Ans.  83  yd. 

8.  Multiply  7  lb  8  ?  5  3  1  3   10  gr.  by  7. 

Ans.  54  ft)  0§  6  3  1  3  10  gr. 

9.  Multiply  24  bu.  1  pk,  6  qt.  by  10. 

10.  Multiply  9  cu.  yd.  15  cu.  ft.  520  cu.  in.  by  5. 

A?is.  ¥1  cu.  yd.  22  cu.  ft.  872  cu.  in. 

11.  Multiply  £84  12s.  6d.  2  far.  by  9. 

12.  If  a  pipe  discharge  4  hhd.  20  gal.  3  qt.  of  water  in 

1  hour,  how  much  will  it  discharge  in  5  hours,  at  the 
same  rate  ?  Ans.  21  hhd.  40  gal.  3  qt. 

13.  If  a  load  of  coal  by  the  long  ton  weigh  1  T.  4  cwt. 

2  qr.  20  lb.,  what  will  be  the  weight  of  6  loads  ? 

Ans.  7  T.  8  cwt.  8  lb. 

14.  If  1  acre  of  land  produce  26  bu.  3  pk.  4  qt.  of 
wheat,  how  much  will  11  acres  produce? 

15.  If  a  man  travel  30  mi.  180  rd.  in  1  day,  how  far 
will  he  travel  in  9  days,  at  the  same  rate  ? 

16.  What  is  the  weight  of  3  dozen  silver  spoons,  each 
dozen  weighing  2  lb.  10  oz.  12  pwt.  14  gr.  ? 

Ans.  8  lb.  7  oz.  17  pwt.  18  gr. 

17.  If  a  wood  chopper  can  cut  2  Cd.  6  cd.  ft.  8  cu.  ft.  of 
wood  in  a  day,  how  many  cords  can  he  cut  in  10  days  ? 

18.  In  20  barrels  of  potatoes,  each  containing  2  bu.  3 
pk.  6  qt.,  how  many  bushels  ?  Ans.  58  bu.  3  pk. 

19.  A  grocer  bought  14  barrels  of  sugar,  each  weighing 
6  cwt.  40  lb. ;  how  much  did  the  whole  weigh  ? 

20.  If  the  sun,  on  an  average,  change  his  latitude  59' 
9"  each  day,  what  will  be  the  change  in  25  days  ? 

21.  If  1  qt.  1  pt.  3  gi.  of  wine  fill  1  bottle,  how  much 
will  be  required  to  fill  3  dozen  bottles  of  the  same  ca- 
pacity ? 


MULTIPLICATION.  161 

22.  If  a  yard  of  cloth  cost  £2  10s.  9d.,  what  will  18 
yards  cost  ?  A7is.  £45  13s.  6d. 

23.  If  a  person  average  8  hr.  20  min.  40  sec.  of  sleep 
daily,  how  much  will  he  sleep  in  30  days  ? 

Ans.  10  da.  10  hr.  20  min. 

24.  How  many  cords  of  wood  in  8  piles,  each  contain- 
ing 40  cd.  ft.  104  cu.  ft.  432  cu.  in..? 

Ans.  46  Cd.  4  cd.  ft.  2  cu.  ft. 

25.  If  each  silver  tahle-spoon  weigh  1  oz.  12  pwt.  16  gi\, 
what  is  the  weight  of  1  dozen  spoons  ? 

26.  If  the  moon's  average  daily  motion  is  33°  10'  35", 
how  much  of  her  orbit  does  she  traverse  in  21  days  ? 

27.  How  much  land  in  12  lots,  each  containing  2  i^^ 
120  P.  ?  Ajis.  33  A. 

28.  How  many  bushels  of  wheat  in  48  sacks,  each  con- 
taining 165  pounds?  Ans.  132  bu. 

29.  If  a  locomotive  move  196  rd.  in  one  minute,  how 
far  will  it  move  in  one  hour  ?  Ans.  36  mi.  240  rd. 

30.  If  a  family  consume  2  gal.  1  qt.  1  pt.  of  molasses  in 
1  week,  how  much  will  they  consume  in  1  year  ? 

Ans.  1  hhd.  60  gal.  2  qt. 

31.  If  it  take  a  man  5  hr.  42  min.  50  sec.  to  saw  a  cord 
of  wood,  how  long  will  it  take  him  to  saw  16  cords  ? 

Ans.  91  hr.  25  min.  20  sec. 

32.  How  many  bushels  of  apples  can  be  j)ut  into  75 
barrels,  each  barrel  containing  3  bu.  1  pk.  ? 

Ans.  243  bu.  3  pk. 

33.  If  a  man  can  build  3  rd.  12  ft.  10  in.  of  wall  in  1 
day,  how  much  can  he  build  in  10  days  ? 

Ans.  37  rd.  12-1-  ft.  4  in. 

34.  If  a  man  can  mow  2  A.  96  P.  of  grass  in  a  day,  how 
much  can  27  men  mow,  at  the  same  rate  ? 

11 


163  compou:n'd   j^umbees. 


DIVISION. 

ITS.  If  4  acres  of  land  produce  102  bu.  2  pk.  2  qt.  of 
wheat,  how  much  will  1  acre  jDroduce  ? 

OPERATION.        -  Analysis.    One  acre  will  pro- 

bu.     pk.    qt.  pt.  duce  ^  as  much  as  4  acres.     Writ- 

er )  102     3     2  ing  the  divisor  on  the  left  of  the 

~      ^      ~      I  dividend,  we  divide  102  bu.  by  4, 

and  obtain  a  quotient  of  25  bu., 
and  a  remainder  of  2  bu.  Write  the  25  bu.  under  the  denomination 
of  bushels,  and  reduce  the  2  bu.  to  pecks,  making  8  pk.,  and  the  3  pk. 
of  the  dividend  added  makes  11  pk.  Dividing  11  pk.  by  4,  we  obtain 
a  quotient  of  2  pk.,  and  a  remainder  of  3  pk.  ;  writing  the  2  pk.  un- 
der the  order  of  pecks,  we  next  reduce  the  3  pk.  to  quarts,  adding 
the  2  qt.  of  the  dividend,  making  26  qt.,  which  divided  by  4  gives  a 
quotient  of  6  qt.,  and  a  remainder  of  2  qt.  Writing  the  6  qt,  under 
the  order  of  quarts,  and  reducing  the  remainder,  2  qt.,  to  pints,  we 
have  4  pt.,  which  divided  by  4  gives  a  quotient  of  1  pt.,  which  we 
write  under  the  order  of  pints,  and  the  work  is  done. 

OPERATION. 

2.  A  farmer  put  132  bu.    pk. 

bu.  1  pk,  of  apples  into  46  )  132     1(2  bu. 
46   barrels  ;   how  many  ^^ 

bu.    did  he   put  into  1  40 

barrel  ?  4 

161(3pk. 
When  the  divisor  is  large,  138 

we  divide  by  long  division,  no 

as  shown  in  the  operation.  r> 

From  these  examples  we  de-  

rive  the  foUowing  184  ( 4  qt. 

184 

Ans.  2  bu.  3  pk.  4  qt. 


Di  VIS  10  If.  163 

Rule.  I.  Divide  the  liigliest  denomination  as  in  simpU 
nmnbers,  and  each  succeeding  denomination  m  the  same 
manner,  if  there  he  no  remainder. 

II.  If  there  he  a  remainder  after  dividing  any  denomi- 
nation, reduce  it  to  the  next  lower  denomination,  adding  in 
the  giveii  number  of  that  de7iomination,  if  any,  and  divide 
as  before. 

III.  Tlie  several  partial  quotients  will  be  the  quotient 
required. 

Examples  for  Peactice. 
(3.)  (4) 

A.   P.  lb.   oz.  pwt.  gr. 

2 )  95  110  3  )52   4  16  18 

47  135  17   5  12   6 

(5.)  (6.) 

■wk.  da.  h.  min.  sec.  bu.  pk.  qt. 

7)33   5  23  45  10         6)88  ^_4 

4  5  20  32  10  14  3  2 

(7.)  •  (8.) 

ft)   !   3  3  gr.  gal.  qt.  pt. 

5 )  28  9  4  2  5  9 )  376  3  1 

5  9  0  2  17  41  3  1 

(9.)  (10.) 

bhd.  gal.  qt.  pt.  A.    P. 

1^  )9  28  2  0  9 )  129  105 

49  2  1  14   65 

(11-)  (13.) 

mi.   rd.   ft.  in.  lb.  oz.  pwt.  gr. 

7  )217  219  12  6      1 1  )185  1  19  13 
31   31   6  6  16  9  19  23 


X64  COMPOUND     NUMBERS. 

13.  Divide  £185  17s.  6d.  by  8.. 

Ans.  £23  4s.  8d.  1  far. 

14.  Divide  16  lb.  14  oz.  10  dr.  by  6. 

^  Ans.  2  lb.  13  oz. 

15.  Divide  358  A.  57  P.  6  sq.  yd.  2  sq.  ft.  by  7. 

Ans.  51  A.  31  P.  8  sq.  ft. 

16.  Divide  192  bu.  3  pk.  1  qt.  1  pt.  by  9. 

Ans.  21  bu.  1  pk.  5  qt.  1  pt. 

17.  Divide  9  hlid.  28  gal.  2  qt.  by  12. 

A71S.  49  gal.  2  qt.  1  pt. 

18.  Divide  328  yd.  1  ft.  3  in.  by  21. 

Ans.  15  yd.  1  ft.  11  in. 

19.  Divide  36  lb  11  ^  4  3  2  3  7  gr.  by  11. 

A71S.  3  lb  4  5  23  1  3  17gr. 

20.  Divide  16  cwt.  3  qr.  18  lb.,  long  ton  weight,  by  32. 

Ans.  2  qr.  3  lb.  3  oz. 

21.  If  a  steamboat  run  174  mi.  26  rd.  in  14  hours,  how 
far  does  she  run  in  1  hour  ? 

22.  A  farm  containing  322  A.  90  P.  is  to  be  divided 
equally  among  13  persons  ;  how  much  will  each  have? 

Ans.  24  A.  130  P. 

23.  A  cartman  drew  38  cd.  5  cd.  ft.  6  cu.  ft.  of  wood, 
at  30  loads  ;  how  much  did  he  average  per  load  ? 

A?is,  1  Cd.  2  cd.  ft.  5  cu.  ft. 

24.  If  24  barrels  of  flour  cost  £98  16s.,  how  much  will 
1  barrel  cost  ?  Ans.  £4  2s.  4d. 

25.  If  a  vessel  sail  135°  16'  12"  in  27  days,  how  far 
does  she  sail  on  an  average  per  day  ? 

A71S.  5°  40'  36". 

26.  If  3  dozen  spoons  weigh  9  lb.  8  oz.  12  gr.,  how  much 
does  each  spoon  weigh?  Ans.  3  oz.  4  pwt.  11  gr. 


PEOMISCUOUS     EXAMPLES.  165 

Promiscuous  Examples. 

1.  A  farmer  raised  200  bu.  2  pk.  of  barley,  175  bu.  3  pk. 
of  corn,  320  bu.  1  pk.  of  oats,  and  225  bu.  2  pk.  of  rye  ; 
what  was  the  whole  quantity  of  grain  raised  ? 

2.  A  person  having  bought  325  A.  80  P.  of  land,  sold 
150  A.  65  P.  of  it ;  how  much  had  he  remaining  ? 

3.  What  is  the  whole  weight  of  72  hogsheads  of  sugar, 
each  weighing  12  cwt.  75  lb.  ?  Ans.  45  T.  18  cwt. 

4.  If  a  railroad  car  run  148  miles  160  rd.  in  8  hours, 
what  is  the  average  rate  of  speed  per  hour  ? 

5.  A  grocer  having  purchased  98  cwt.  50  lb.  of  sugar, 
sold  10  cwt.  45  lb.  to  one  man,  and  18  cwt.  16  lb.  to  an- 
other ;  how  much  remained  unsold  ? 

6.  Bought  12  tea-spoons,  each  weighing  16  pwt.  20  gr., 
and  6  table-spoons,  each  weighing  1  oz.  12  pwt.  ;  what 
was  their  total  weight?  A7is.  1  lb.  7  oz.  14  pwt. 

7.  A  farmer  raised  24  T.  17  cwt.  of  hay  ;  he  sold  5  loads, 
each  weighing  1  T.  8  cwt.  21  lb. ;  how  much  has  he  re- 
maining ?  A71S.  17  T.  15  cwt.  95  lb. 

8.  A  jeweler  having  36  lb.  10  oz.  14  pwt.  of  silver,  uses 
21  lb.  6  oz.  of  it,  and  then  manufactures  the  remainder 
into  8  tea-pots  ;  what  is  the  weight  of  each  ? 

A71S.  1  lb.  11  oz.  1  pwt.  18  gr. 

9.  A  man  purchasing  2  A.  140  P.  of  land,  reserves  |-  an 
acre  for  his  own  use,  and  divides  the  remainder  in  4  equal 
lots  ;  how  much  does  each  lot  contain?  A7is.  95  P. 

10.  How  many  pounds  of  sugar  in  28  barrels,  each  con- 
taining 3  cwt.  42  lb.?  A}is.  9576  pounds. 

11.  If  from  a  piece  of  land  containing  5  A.  120  P.,  2  A. 
72  P.  be  taken,  how  many  square  rods  will  remain  ? 


166  COMPOUND     K  UMBERS. 

12.  Divide  a  tract  of  land  containing  1299500  square 
rods  into  25  farms  of  equal  area;  how  much  will  there 
be  in  each  ?  Ans.  324  A.  140  P. 

13.  A  merchant  buys  3  hogsheads  of  molasses  at  30 
cents  a  gallon,  and  sells  it  at  45  cents  ;  what  does  he  gain 
on  the  whole  ? 

14.  What  is  the  cost  of  3  chests  of  tea,  each  weighing 
2  cwt.  68  lb.,  at  $.84  a  pound  ?  Ans.  $675.36. 

15.  How  many  steps  of  30  inches  each  must  a  person 
take  in  walking  12  miles  ? 

16.  If  a  man  buy  10  bushels  of  chestnuts,  at  $3  a  bushel, 
and  sell  them  at  10  cents  a  pint,  what  is  his  whole  gain  ? 

A71S.  $34. 

17.  How  many  times  will  a  wheel  13  ft.  4  inches  in 
circumference  turn  round  in  going  12  miles  ? 

Ans.  4752. 

18.  If  8*  horses  eat  12  bu.  3  pk.  of  oats  in  3  days,  how 
many  bushels  will  20  horses  eat  in  the  same  time  ? 

Ans.  31  bu.  3  pk.  4  qt. 

19.  How  much  sugar  at  9  cents  a  pound  must  be  given 
for  2  cwt.  43  lb.  of  pork  at  6  cents  a  pound  ? 

Ans.  162  pounds. 

20.  How  many  cubic  feet  in  a  room  18  feet  long,  16  feet 
wide,  and  10  feet  high  ? 

21.  A  person  wishes  to  ship  720  bushels  of  potatoes  in 
barrels,  which  shall  hold  3  bu.  3  pk.  each ;  how  many 
barrels  must  he  use?  Ans.  192. 

22.  How  many  rods  of  fence  mil  inclose  a  farm  1  mile 
square  ?  Ans.  1280  rods. 

23.  If  granite  weigh  175  pounds  a  cubic  foot,  what  is 
the  weight  of  a  cubic  yard  ? 

Ans.  2  T.  7  cwt.  25  lb. 


OANCELLATIOK.  167 


V    --s,- 


OANCELLATIOK. 

174.  Cancellation  is  the  process  of  rejecting  equal 
factors  from  numbers  sustaining  to  each,  other  the  rela- 
tion of  dividend  and  divisor. 

It  lias  been  shown  ( 70  )  that  the  dividend  is  equal  to 
the  product  of  the  divisor  multiplied  by  the  quotient. 
Hence,  if  the  dividend  can  be  resolved  into  two  factors, 
one  of  which  is  the  divisor,  the  other  factor  will  be  the 
quotient. 

1.  Divide  72  by  9. 

OPERATION.  Analysis.     We  see  in  this 

Divisor.    0)0x8  Dividend.  example,  tliat  72   is   composed 

^  .  of  tlie  factors  9  and  8,  and  that 

'  tlie  factor  9    is    equal  to  tlie 

divisor.     Therefore  we  reject  the  factor  9,  and  the  remaining  factor, 

8,  is  the  quotient. 

174.  ^Tienever  the  di^-idend  and  divisor  are  each 
composite  numbers,  the  factors  common  to  both  may  first 
be  rejected  without  altering  the  final  result. 

2.  What  is  the  quotient  of  48  divided  by  24  ? 

OPEEATiON.  Analysis.    We  first  indicate 

48      $  X  6  X  2  ^^  operation  to  be  performed 

24~~S~y~i         -—^f  A71S.  i3y  -vtaiting  the  dividend  above 

a  line,  and  the  divisor  below  it. 
We  resolve  48  into  the  factors  3,  8,  and  2,  and  24  into  the  factors  3, 
and  8.  We  next  cancel  the  factors  3,  and  8,  which  are  common  to 
the  dividend  and  divisor,  and  we  have  left  the  factor  2,  in  the 
dividend,  which  is  the  quotient. 

When  all  the  factors  or  numbers  in  the  dividend  are  cancelled,  1  should  he 
retained. 


/ 


168  CANCELLATION". 

1T5.  If  any  two  numbers,  one  in  the  dividend  and  one 
in  the  divisor,  contain  a  common  factor,  we  may  reject 
that  factor. 

3.  In  15  times  63,  how  many  times  45  ? 

OPEEATION.  A]Sf  ALYsis.     Ill  this  example  we  see 

^%      A  tliat  5  will  divide  15  and  45 ;  so  we 

■  -  =  21    AnS,         reject  5  as  a  factor  of  15,  and  retain 

^$  the  factor  3,  and  also  as  a  factor  of  45, 

^  and  retain  the  factor  9.     Again  9  will 

divide  9  in  the  divisor,  and  63  in  the 
dividend.  Dividing  both  numbers  by  9,  1  \vill  be  retained  in  the 
divisor,  and  7  in  the  dividend.  Finally  the  product  of  3  x  7=21,  the 
quotient. 

4.  What  is  the  quotient  of  25  x  18  x  6  x  4,  divided  by 
15x4x9x3? 

OPERATION. 

A        A         ^         y        ^        ^        ^        .^r.  Als^ALYSIS.   In 

X$X    ix$X$  3  3  preceding    ex- 

3  ample,  we  re- 

ject all  the  factors  that  are  common  to  both  dividend  and  divisor, 
and  have  remaining  the  factor  3  in  the  divisor,  and  the  factors  5,  2, 
and  2  in  the  di\ddend.     Completing  the  work,  xie  have  ^-=6^,  Ans. 

From  the  preceding  examples  and  iUustrations  we  de- 
rive the  following 

Rule.  I.  Write  the  mimhers  composing  the  dividend 
above  a  horizontal  line,  and  the  numbers  composing  the  di- 
visor beloio  it. 

II.  Cancel  all  the  factors  common  to  loth  dividend  and 
divisor. 

III.  Divide  the  product  of  the  remaining  factors  of  the 
dividend  by  the  product  of  the  remaining  factors  of  the 
divisor,  and  the  result  tvill  be  the  quotie7it. 


CAITCELLATION. 


169 


1.  RejectiDg  a  factor  from  any  number  is  dmding  the  number  by  that  factor. 

2.  When  a  factor  is  canceled,  the  unit,  1,  is  supposed  to  take  its  place. 

3.  One  factor  in  the  dividend  will  cancel  only  one  equal  factor  in  the  divisor. 

4.  If  all  the  factors  or  numbers  of  the  divisor  are  canceled,  the  product  of  the 
remaining  factors  of  the  dividend  will  be  the  quotient. 

5.  By  many  it  is  thought  more  convenient  to  write  the  factors  of  the  dividend 
on  the  right  of  a  vertical  line,  and  the  factors  of  the  divisor  on  the  left. 

Examples  foe  Peactice. 

1.  Divide  tlie  product  of  12  x  8  x  6  by  8  x  4  x  3. 

FIRST  OPEEATION.  SECOND  OPERATION. 

l^;^x$xl    3x2    ^    , 
ex.4x3        1 

6,  Ans. 

2.  Divide  the  product  of  25  x  18  x  4  x  3  by  7  x  6  x  5  x  3. 

FIRST   OPERATION. 

?pxiax4xS     5x3x4     60     ^,      , 

-= — =84,  A?is, 


$ 

1^3 

4 

a 

$ 

62 

7x  0  x?x$ 


7 


rv 

i 


SECOND  OPERATION. 


7 

205 

0 

W 

$ 

4 

$ 

$ 

7 

60 

8^-,  Ans. 

3.  Di\ide  tlie  product  of  36  x  10  x  7  by  14  x  5  x  9. 

Ans.  4. 

4.  What  is  the  quotient  of  21x8x40x3  di^-ided  by 
12x7x20?  Ans.  12.   ' 

5.  What  is  the  quotient  of  64  x  1 8  x  9  didded  by  30  x 


27x4? 


Ans.  34-. 


6.  Divide  the  product  of  120  x  44  x  6  by  60  x  11  x  8. 

A}is.  6. 


170  CAK'CELLATION'. 

7.  Multiply  200  by  GO,  and  divide  the  product  by  50 
multiplipd  by  4^8.  Arts.  5. 

8.  Multiply  8  times  32  by  6  times  27,  and  divide  the 
product  by  9  times  96.  Aiis.  48. 

9.  What  is  the  quotient  of  21  x  8  x  60  x  8  x  6  divided  by 
7x12x3x8x3?  ^?^5.  80. 

10.  What  is  the  quotient  of  18x6x4x42  divided  by 
4x9x3x7x6?  Ans.  4. 

11.  If  18  X  5  X  9  X  m  be  divided  by  40  x  22  x  6,  what  is 
the  quotient?  Ajis.  lOf. 

12.  The  product  of  the  numbers  26,  11,  and  21,  is  to  be 
divided  by  the  product  of  the  numbers  14  and  13  ;  what 
is  the  quotient  ?  Ajis.  33. 

13.  The  product  of  the  numbers  48,  72,  28,  and  5,  is  to 
be  divided  by  the  product  of  the  numbers  84,  15,  7,  and 
6  ;  what  is  the  quotient  ?  A7is.  9^. 

14.  How  many  tons  of  hay  at  19  a  ton,  must  be  given 
for  27  cords  of  wood,  at  14  a  cord  ?  A71S.  12  tons. 

15.  How  many  bushels  of  corn,  worth  60  cents  a  bushel, 
must  be  given  for  25  bushels  of  rye,  worth  90  cents  a 
bushel?  ^;25.  3 7-|- bushels. 

16.  How  many  peaches  worth  2  cents  each  must  be  given 
for  48  oranges,  at  3  cents  each  ?  Ans.  72, 

17.  How  many  days'  work,  at  75  cents  a  day,  will  pay  for 
30  pounds  of  coffee,  at  15  cents  a  pound  ?    Ans.  6  days. 

18.  How  many  suits  of  clothes,  at  $18  a  suit,  can  be  made 
from  5  pieces  of  cloth,  each  piece  containing  24  yards,  at 
$3  a  yard  ?  Ans.  20  suits. 

19.  How  many  tubs  of  butter,  each  containing  48 
pounds,  at  14  cents  a  pound,  must  be  given  for  3  boxes  of 
tea,  each  containing  42  pounds,  worth  60  cents  a  pound  ? 

Ans.  11  J. 


CANCELLATIOlN.  17l 

20.  iIoTV  many  days'  work,  at  84  cents  a  day,  will  pay 
for  36  bushels  of  corn  worth  56  cents  a  bushel  i" 

A71S.  24. 

21.  A  farmer  exchanged  45  bushels  of  potatoes  worth 
30  cents  a  bushel,  for  15  pounds  of  tea  ;  what  was  the  tea 
worth  a  pound?  Ans.  90  cents. 

22.  A  grocer  bought  120  pounds  of  cheese,  at  9  cents  a 
pound,  and  paid  in  molasses,  at  45  cents  a  gallon ;  how 
many  gallons  of  molasses  paid  for  the  cheese  ? 

Ans.  24  gallons. 

23.  Grave  12  barrels  of  flour,  at  $7  a  barrel,  for  hay 
worth  $18  a  ton  ;  how  many  tons  of  hay  was  the  flour 
worth  ?  Ans.  4f  tons. 

24.  Sold  8  firkins  of  butter,  each  weighing  56  pounds, 
at  15  cents  a  pound,  and  received  in  payment  3  boxes  of 
tea,  each  containing  40  pounds  ;  what  was  the  tea  worth 
a  pound?  Ans.  56  cents. 

25.  A  man  took  6  loads  of  apples  to  market,  each  load 
containing  14  barrels,  and  each  barrel  3  bushels.  He  sold 
them  at  50  cents  a  bushel,  and  received  in  payment  9  bar- 
rels of  sugar,  each  weighing  210  pounds  ;  what  was  the 
sugar  worth  a  pound  ?  Afis.  6f  cents. 

26.  A  grocer  sold  12  boxes  of  soap,  each  containing  51 
pounds,  at  10  cents  a  pound  ;  he  received  in  payment  a 
certain  number  of  barrels  of  potatoes,  each  containing  3 
bushels,  at  30  cents  a  bushel ;  how  many  barrels  did  he 
receive?  A7is.  68  barrels. 

27.  A  man  sold  4  loads  of  barley,  each  load  containing 
50  bushels,  at  70  cents  a  bushel,  and  received  in  payment 
I  pieces  of  cloth,  each  piece  containing  35  j^ards  ;  what 
Rras  the  cloth  worth  a  yard?  Ans.  12.40. 


173  ANALYSIS. 

AISTALYSIS. 

176,  Analysis,  in  arithmetic,  is  the  process  of  solving 
problems  independently  of  set  rules,  by  tracing  the  rela- 
tions of  the  given  numbers  and  the  reasons  of  the  separate 
steps  of  the  operation  according  to  the  special  conditions 
of  each  question. 

11"7.  In  solving  questions  by  analysis,  we  generally 
reason  from  the  give7i  mmibev  to  unity,  or  1,  and  then 
from  unity,  or  1,  to  the  required  num'ber. 

178.  United  States  money  is  reckoned  in  dollars,  dimes, 

cents,  and  mills,  one  dollar  being  uniformly  valued  in  all 

the  States  at  100  cents. 

At  the  time  of  the  adoption  of  our  decimal  currency  by  Congress, 
in  1 786,  the  colonial  currency,  or  Mils  of  credit,  issued  by  the  colo- 
nies, had  depreciated  in  value;  and  this  depreciation,  being  unequal 
in  the  different  colonies,  gave  rise  to  the  different  values  of  the 

State  currencies. 

Georgia  Currency. 
Georgia,  South  Carolina $l=4s.  8d.=56d. 

Canada  Currency. 
The  Dominion  of  Canada $l=:os.=:60d. 

New  England  Currency. 
New  England,  Indiana,  Illinois,  Missouri,  Vir- )  ^^  _„    _„f. , 

ginia,  Kentucky,  Tennessee,  Mississippi,  Texas.)  5S>1— OS.— /^d. 

Pennsylvania  Currency. 
New  Jersey,  Pennsylvania,  Delaware,  Maryland.  .$l=7s.  6d.=90d, 

JSfew  York  Currency. 
New  York,  Ohio,  Michigan,  North  Carolina $l=8s.=:96d. 

In  many  of  the  States  it  was  customary  to  give  the  retail  price  of 
articles  in  shillings  and  pence,  and  the  cost  of  the  wliole  in  dollars 
and  cents. 

This  usage  has  become  nearly  if  not  quite  obsolete  ali  over  the 
country ;  but  the  matter  has  an  historical  interest,  and  is  retained  in 
this  new  edition,  to  avoid  derangement  with  previous  editions,  and 
the  examples  afford  a  pleasant  and  profitable  exercise  for  the  pupiL 
It  may  be  omitted  however,  at  the  discretion  of  the  teacher. 


ANALYSIS. 


173 


The  following  will  be  found  an  easy,  short,  h.nd  prac- 
tical method  of  reducing  currencies  to  dollars  and  cents  : 


Examples  for  Practice. 

1.  What  will  be  the  cost  of  36  bushels  of  apples,  at  3 
shillings  a  bushel,  New  England  Currency  ? 


OPERATION. 


36x3::=108s. 
108 -^6  =  $18.  Or,  $ 


$0 
3 


118,  Ans. 


Analysis.  Since  1 
bushel  costs  3  shillings, 
36  bushels  will  cost  36 
times  3s.,  or  36x3=108s. ; 
and  as  6s.  make  1  dollar, 
New    England    currency, 


there  are  as  many  dollars  in  108s.  as  six  is  contained  times  in  108, 
or  108-r-6=:$18. 

2.  What  will  112  bushels  of  barley  cost,  at  5s.  6d.  per 
bushel,  New  York  currency  ? 


OPERATION. 


Or, 


177 


11 


$77,  Ans. 


Analysis.  Multiply 
the  number  of  bushels 
by  the  price,  and  di- 
vide the  result  by  the 
value  of  1  dollar  as  in 
the  first  example,  re- 
ducing both  the  price 
Or,  when  the  price  is  an 


and  1  dollar  to  pence,  and  we  obtain  $77 
aliquot  part  of  a  shilling,  the  price  may  be  reduced  to  an  improper 
fraction  for  a  multiplier,  thus:  53.  6d.=5|s.=-V-s.,  the  multiplier. 
The  value  of  a  dollar  being  8s.,  divide  by  8  as  in  the  operation. 
Hence,  to  find  the  cost  of  an  article  in  dollars  and  cents,  when  the 
price  is  in  shillings  and  pence, 

MuUiphj  the  commodity  ly  the  price,  and  divide  the  prO' 
duct  ly  the  value  of  one  dollar  in  the  required  currency, 
reduced  to  the  same  denominational  unit  as  the  price* 


174 


ANALYSIS. 


3.  What  will  180  cords  of  wood  cost  at  8s.  4d.  per  cord, 
Pennsylvania  currency  ? 


X$0 
100 


$200 


OPERATION. 

Or, 

i4 

$ 

25 

t$ 

2 

$200,  Ans, 


Analysis.     Multiply  the 
quantity    by    the    price    in 
pence,  and  divide  the  product 
by  the  value  of  1  dollar  in 
pence ;'  or,  reduce  the  shil- 
lings an4'  pence,  both  of  the 
price   and   of  the  dollar,  to 
the  fraction  of  a  shilling  before  multiplying  and  dividing,  thus  : 
8s.  4d.=:84^s.=^s.,  the  multiplier.     The  value  of  the  dollar  being 
7fl.  6d.=:7|s.=^s.,  we  divide  by  ^-,  as  in  the  operation. 

4.  What  will  be  the  cost  of  7-|-  yards  of  cloth,  at  6s.  8d. 
Kew  York  currency  ? 

OPERATION. 

g 

Analysis.  We  reduce  the  quantity 
and  the  price  to  improper  fractions,  be- 
fore multiplying. 

$6.25,  Ans, 

When  there  is  a  remainder  in  the  dividend,  it  may  be  reduced  to  cents  an<3 
mills  by  annexing  two  or  three  ciphers  and  continuing  the  division. 

5.  What  will  7  hhd.  of  molasses  cost,  at  Is.  3d.  per 
quart,  Georgia  currency  ? 

Analysis.  In  this  example  we 
first  reduce  7  nlid.  to  quarts,  by  mub 
tiplying  by  63,  and  4,  and  then  mul, 
tiply  by  the  price,  either  reduced  to 
pence  or  to  an  improper  fraction,  and 
divide  by  the  value  of  1  dollar  re, 
duced  to  the  same  denomination  aa 
$472.50,  Ans,        the  price. 


2 

U, 

$ 

n 

^$ 

4 

25.00 

OPERATION. 

t 

63 

2 

^ 

H 

15 

2 

945.00 

AI^ALYSIS.  175 

6.  Sold  8  firkins  of  butter,  each  containing  56  pounds, 
at  Is.  3d.  per  pound,  and  received  in  payment  tea  at 
6s.  8d.  per  pound ;  how  many  pounds  of  tea  would  pay 
for  the  butter  ? 

OPERATION.  Analysis.     The  operation  in  tliis 

^              $  is  similar  to  tlie  preceding  examples, 

10         502  8  except  tiiat  we  divide  the  cost  of  the 

^0    103  "butter  by  the  price  of  a  unit  of  the 

Ans,  84  pounds.      ^''\f'  '"''''^'f  ^^  P">'^?^^^;  ^^^^^^^ 

to  the  same  denominational  unit  as 

the  price  of  a  unit  of  the  article  sold.     The  result  will  be  the  same 

in  whatever  currency. 

7.  What  will  be  the  cost  of  a  load  of  oats  containing  64 
bushels,  at  2s.  6d.  a  bushel,  New  York  currency  ? 

Ans.  $20. 

8.  At  9d.  a  pound,  Avhat  will  be  the  cost  of  120  pounds 
of  sugar,  New  England  currency  ?  Ans.  ^15. 

9.  What  will  be  the  value  of  a  load  of  potatoes  measur- 
ing 35  bushels,  at  2s.  3d.  a  bushel,  Penn.  currency  ? 

Ans.  $10.50. 

10.  What  'R^ll  be  the  cost  of  240  bushels  of  wheat,  at 
9s.  4cl.  a  bushel,  Michigan  currency?  Ans.  $280. 

11.  In  New  Jersey  currency  ?  Ans.  $298. 66|. 

12.  In  Illinois  currency?  Ans.  $373.33-1-. 

13.  In  South  Carolina  currency?  Ans.  $480. 

14.  In  Virginia  currency  ?  A^is. 

15.  In  Ohio  currency?  Ans. 

16.  In  Canada  currency?  Afis.  $448. 

17.  How  many  days  work  at  7s.  6d.  ada}^,  must  be  given 
for  5  bushels  of  wheat  at  10s.  a  bushel  ?    A?is.  6f  days. 

18.  What  will  be  the  cost  of  5  casks  of  rice,  each 
weighing  168  pounds,  at  3d.  per  pound,  South  Carolina 
currencv?  Ans.  $45. 


176  ANALYSIS. 

^^9.  How  many  pounds  of  sugar,  at  9d.  per  pound,  must 
be  given  for  18  bushels  of  apples,  at  2s.  7d.  per  bushel  ? 

A?is.  62  pounds. 

20.  Bought  3  casks  of  catawba  wine,  each  cask  contain- 
ing 64  gallons,  at  7s.  9d.  per  quart,  Ohio  currency  ;  what 
was  the  cost  of  the  whole?  Ans.  1744. 

21.  What  will  it  cost  to  build  150  rods  of  wall,  at  3s.  8d. 
per  rod,  Canada  currency?  Ans,  $110. 

22.  How  many  pounds  of  butter,  at  18d.  a  pound,  must 
be  given  for  12  j)ounds  of  tea,  at  6s.  4d.  a  pound? 

Ans,  42f  pounds. 

23.  What  will  be  the  cost  of  4  hogsheads  of  molasses, 
at  Is.  2d.  per  quart,  Mississippi  currency?     Ans.  $196. 

24.  A  farmer  exchanged  28  bushels  of  barley,  worth 
5s.  8d.  a  bushel,  with  his  neighbor,  for  corn  worth  7s.  a 
bushel ;  how  many  bushels  of  corn  was  the  barley  worth? 

Ans.  22f  bushels. 

25.  What  will  a  load  of  wheat,  measuring  45  bushels, 
be  worth  at  lis.  a  bushel,  Kentucky  currency  ? 

A72S.  182.50. 

26.  What  will  12  yards  of  Irish  linen  cost,  at  4s.  9d.  a 
yard,  Pennsylvania  currency?  Ans.  17.60. 

27.  Bought  the  following  bill  of  goods  of  Trade  well  & 
Co. ;  how  much  did  the  whole  amount  to,  New  York 
currency? 


4  yards  of  cloth, 

at 

5s.  6d.  per  yard. 

9      "         calico,  - 

{( 

Is.  4d. 

10      ''         ribbon,    - 

a 

2s.  3d. 

6  gallons  molasses. 

a 

4s.  8d.  per  gallon. 

3|  pounds  of  tea^     - 

<< 

6s.  per  pound. 
Ans.  $13.1875. 

PERCENTAGE.  177 


PEECEjSTTAGE. 

'^  IW,  Per  Cent,  is  a  term  derived  from  tlie  Latin  words 
per  centum,  and  signifies  hy  the  hundred,  or  hundredths, 
that  is^  a  certain  number  of  parts  of  each  one  hundred 
parts,  of  whatever  denomination.  Thus,  by  5  per  cent, 
is  meant  5  cents  of  every  100  cents,  $5  of  every  $100,  5 
bushels  of  every  100  bushels,  etc.  Therefore,  5  per  cent, 
equals  5  hundredths^. 05  =j|-5-:i::-^^^.  8  per  cent,  equals 
8  hundredths=.08=y|-o— -^. 

180o  Percentag-e  is  such  a  part  of  a  number  as  is  in- 
dicated by  the  per  cent. 

181.  The  Base  of  percentage  is  the  number  on  which 
the  percentage  is'computed. 

182.  Since  per  cent,  is  any  number  of  hundredths,  it 
is  usually  expressed  in  the  form  of  a  decimal  or  a  common 
fraction,  as  in  the  following 

Table. 

Decimals.    Common  Fractions.    Lowest  Tenns. 
1  per  cent.        —  .01        =        ^^        =         -^^ 


2  —  JL 

ToU  —  50 

4  — .                  1 

TI77  —  a^ 


3  per  cent.  =  .02 

4  per  cent.  =  .04 

5  per  cent.  =  .05  =  y^-j  =  4^ 

6  per  cent.  =  .06  =  jf^  =  5% 

7  per  cent.  =  .07  =z  ^^  =  -^-^ 

8  per  cent.  =  .08  =  ifg-  =  ai" 
10  per  cent.  =  .10  =  -j^o  =  xV 
16  per  cent.  =  .16  =  ^i^  =  -i^ 
20  per  cent.  =  .20  =  ^  =  \ 
25  per  cent.  =  .25  =  -f-^^  =  ^ 
50  per  cent.  =  .50  =  -^^-^  =  ^ 

100  per  cent.  =  1.00  ==  {^  ^  1 

13 


178  PERCENTAGE. 

183.  To  find  the  percentage  of  any  number. 

I.  A  man  having  $120,  paid  out  5  per  cent,  of  it  for 
groceries  ;  how  much  did  he  pay  out  ? 

OPERATION. 

^^-^^  Analysis.     Since  5  per  cent,  is  y|-^=.05i  he 

•^^  paid  out  .05  of  $120,  or  $120  x  .05=i$6. 

$6.00 

Rule.  Multiply  the  given  numher  or  quantity  hy  the 
rate  per  cent.y  expressed  decimally,  and  point  off  as  in 
decimals* 

Examples  for  Practice. 

2.  What  is  4  per  cent,  of  1300  ?  Ans.  $12. 

3.  What  is  3  per  cent,  of  1175  ?         '      Ans.  $5.25. 

4.  What  is  5  per  cent,  of  450  jDOunds  ? 

5.  What  is  6  per  cent,  of  65  gallons?    Ans.  3.9  gal. 

6.  What  is  9  per  cent,  of  200  sheep  ?  Ans.  18  sheep. 

7.  What  is  7  per  cent,  of  $97?  Ans.  $6.79. 

8.  AYhat  is  10  per  cent,  of  $12.50  ?  Ans.  $1.25. 

9.  What  is  40  per  cent,  of  840  men  ?  A7is.  336  men. 
10.  What  is  25  per  cent,  of  740  miles  ? 

II.  A  man  having  $4000,  invests  25  per  cent,  of  it  in 
land  ;  what  sum  does  he  invest  ?  Ans.  $1000. 

12.  A  man  bought  1500  barrels  of  apples,  and  found  on 
opening  them  that  12  per  cent,  of  them  were  spoiled  ; 
how  many  barrels  did  he  lose  ?  Ans.  180  barrels. 

13.  A  farmer  having  180  sheep,  sold  45  per  cent,  of 
them  and  kept  the  remainder ;  how  many  did  he  sell  and 
how  many  did  he  keep  ?  Ans.  He  kept  99. 

14.  Having  deposited  $1275  in  bank,  I  draw  out  8  per 
cent,  of  it  ;  how  much  remains  ?  Ans,  $1173. 


COMMISSION".  179 


COMMISSION. 


184.  An  Agent,  Factor,  or  Broker  is  a  person  wlio 
transacts  business  for  another. 

185.  A  Conimission  Merchant  is  an  agent  who  buys 
and  sells  goods  for  another. 

186.  Commission  is  the  fee  or  compensation  of  an 
agent,  factor,  or  commission  merchant. 

J  87.  To  find  the  commission  or  brokerage  on  any 

sum  of  money. 

1.  A  commission  merchant  sells  butter  and  cheese  to  the 
amount  of  $1540  ;  what  is  his  commission  at  5  per  cent.  ? 

OPERATION.  Analysis.    Since  tlie  commission 

$1540  X  .05  =  $77,  Ans.      on  $1  i^  5  cents,  or  .05  of  a  dollar, 

on  $1540  it  is  $1540  x  .05=$77. 

KuLE.  Multiply  the  given  sum  hy  the  rate  per  cent., 
expressed  decimally  ;  the  result  will  he  the  commission  or 
hrokerage. 

Examples  for  Practice. 

2.  What  commission  must  be  paid  for  collecting  S3  840, 
at  3  per  cent.  ?  Ans.  $115.20. 

3.  A  commission  merchant  sells  goods  to  the  amount 
of  $5487.50  ;  what  is  his  commission  at  2  per  cent.  ? 

A71S.  S109.75. 

4.  An  agent  buys  5460  bushels  of  wheat  at  $1.50  a 
bushel ;  how  much  is  his  commission  for  bujang,  at  4  per 
cent.?  A71S.  $327.60. 

5.  A  commission  merchant  sells  400  barrels  of  potatoes 
at  $2.25  a  barrel,  and  345  barrels  of  apples  at  $3.20  a  barrel; 
how  much  is  his  commission  for  selling,  at  5  per  cent.  ? 

6.  An  agent  sold  my  house  and  lot  for  $6525 ;  what  was 
his  commission  at  2  per  cent.  ? 


180  PEBCENTAGE. 

PROFIT    AND    LOSS. 

188.  Profit  and  Loss  are  commercial  terms,  used  to 
express  the  gain  or  loss  in  business  transactions,  wliich  is 
usually  reckoned  at  a  certain  per  cent,  on  the  prime  or 
first  cost  of  articles. 

189.  To  find  the  amount  of  profit  or  loss,  when  the 
cost  and  the  gain  or  loss  per  cent,  are  given. 

1.  A  man  bought  a  horse  for  $135,  and  afterward  sold 
him  for  20  per  cent,  more  than  he  gave ;  how  much  did 
he  gain  ? 

OPERATION.  Analysis.       Since   $1    gains  20 

1135  X. 20  =  127,  Ans.      cents,    or  20    per  cent.,    $185   will 

gain  $135  x  .20=$27. 

KuLE.  Multiply  the  cost  hy  the  rate  per  cent.,  expressed 
decimally. 

Examples  for  Practice. 

2.  Bought  a  horse  for  $150,  and  sold  him  at  15  per 
cent,  profit ;  what  was  my  gain  ?  Ans.  $22.50. 

3.  Bought  25  cords  of  wood  at  13.50  a  cord,  and  sold 
it  so  as  to  gain  33  per  cent. ;  what  did  I  make  ? 

Ans.  $28,871 

4.  Paid  7  cents  a  pound  for  2480  pounds  of  pork,  and 
afterward  lost  10  per  cent,  on  the  cost,  in  selUng  it ;  what 
was  my  whole  loss?  Ans.  $17.36. 

5.  Bought  1000  bushels  of  wheat  at  $1.25  a  bushel, 
and  sold  the  flour  at  18  per  cent,  advance  on  the  cost  of 
the  wheat ;  what  was  my  whole  gain  ?  Ans.  $225. 

8.  A  grocer  bought  6  barrels  of  sugar,  each  containing 
220  pounds,  at  7|-  cents  a  pound,  and  sold  it  at  20  per 
cent,  profit ;  what  was  the  whole  gain?      Ans.  $19.80. 


SIMPLE     INTEREST. 


181 


SIMPLE    INTEEEST. 

190.  Interest  is  a  sum  paid  for  tlie  use  of  money. 

191.  Principal  is  the  sum  for  the  use  of  which  inter- 
est is  paid. 

193.  Rate  per  cent,  per  annum  is  the  sum  per  cent 
paid  for  the  use  of  $100  annually. 

The  rate  per  cent,  is  commonly  expressed  decimally,  as  hundredths  (182). 

193.  Amount  is  the  sum  of  the  principal  and  in- 
terest. 

194.  Simple  Interest  is  the  sum  paid  for  the  use  of 

the  principal  only,  during  the  whole  time  of  the  loan  or 
credit. 

195.  ILeg-al  Interest  is  the  rate  per  cent,  established 
by  law.     It  varies  in  different  States  as  follows  : 


Alabama 8  per  cent. 

Arkansas 6  " 

California ....10  " 

Connecticat 7  " 

Delaware 6  " 

Dist.  of  Columbia...  6  " 

Florida 8  " 

Georgia 7  " 

Illinois C  " 

Indiana* G  " 

Iowa ^ 6  *' 

Kentucky 6  " 

Louisiana 5  *' 

Maine 6  " 

Maryland 6  " 

Massacliusetts 6  " 

Micliigan 7  " 


Minnesota 7 

Mississippi 6 

Missouri 6 

New  Hampsliire 6 

New  Jersey 6 

New  York.   6 

North  Carolina 6 

Oliio G 

Pennsylvania 6 

Rhode  Island G 

Soutli  Carolina 7 

Tennessee G 

Texas 8 

U.  S.  (debts) G 

Vermont G 

Virginia G 

Wisconsin 7 


per  cent. 


r< 
ii 
<< 
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<c 

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<l 

<  ( 

<  ( 
<( 
<< 

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(( 
l( 
<< 

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t( 
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l< 
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(( 
(( 


1.  The  le.2;al  rate  in  Canada,  Nova  Scotia,  and  Ireland  is  6  per  cent.,  and  in 
England  and  France  5  per  cent. 

2.  When  the  rate  per  cent,  is  not  specified  In  accounts,  notes,  mortgages,  con- 
tracts, etc,  the  legal  rate  is  always  understood. 


182  PEECEKTAGE. 

Case  I. 

196.  To  find  the  ioterest  on  any  sum,* at  any  rate 
per  cent.,  for  years  and  months. 

1.  What  is  the  interest  on  $140  for  3  years  3  months, 
at  7  per  cent.  ? 

OPERATION. 

$140  ■    * 

.07  ANALYSIS.      Tlie    interest    on 

$140,  for  1  yr.,  at  7  per  cent.,  is 
.07  of  the  principal,  or  $9.80,  and 


19.80   Int.forlyr. 


31 

ji-  the  interest  for  3  yr.  3mo.  is  3j%= 


245  8 J  times  the  interest  for  one  yr., 

2940  or  $9.80  x  3^,  wliich  is  $31.85. 


AjIS.    $31.85    Tut.  for  3 yr.  3  mo. 

Rule.  I.  3fuUipIi/  the  2J'i^'^^cipal  hy  tlie  rate  per  cent., 
and  tlie  product  will  he  tlie  interest  for  1  year. 

II.  Multiply  this  product  hy  the  ii?ne  in  years  and  frac- 
tions of  a  year,  and  the  result  will  he  the  required  interest. 

Examples  for  Peactice. 

2.  What  is  the  interest  on  $48.50  for  2  years  6  months, 
at  6  per  cent.  ?  Ans.  $7,275. 

3.  What  is  the  interest  on  $325.41  for  3  years  4  months, 
at  5  per  cent.?  Ans.  $54,235. 

4.  What  is  the  interest  on  $279.60  for  1  year  ^months, 
at  7  per  cent.  ?  Ans,  $34,251. 

5.  What  is  the  amount  of  $26.84  for  2  yr.  6  mo.,  at  5 
per  cent.  ?  A-ns.  $30,195. 

6.  What  is  the  amount  of  $20©  for  1  yr.  9  mo.,  at  7 
percent?  Ans.  $224.50. 

7.  What  is  the  interest  on  $750  for  1  year  3  months, 
at  5  per  cent?  Ans,  $46,875, 


simple    interest.  183 

•  Case  II. 

197.  To  find  the  interest  on  any  sum,  for  any  time, 
at  any  rate  per  cent. 

Obvious  Relations  between  Time  and  Interest. 

I.  The  interest  on  any  sum  for  1  year,  at  1  per  cent.,  is 
.01  of  that  sum,  and  is  equal  to  the  principal  with  the 
separatrix  removed  two  places  to  the  left. 

II.  A  month  heing  ^  of  a  year,  -^^  ^^  ^^^  interest  on 
any  sum  for  1  year  is  the  interest  for  1  month. 

III.  The  interest  on  any  sum  for  3  days  is  -^=z^^=:.l 
of  the  interest  for  1  month,  and  any  number  of  days  may 
readily  be  reduced  to  tenths  of  a  month  by  dividing  by  3. 

IV.  The  interest  on  any  sum  for  1  month,  multiplied 
by  any  given  time  expressed  in  months  and  tenths  of  a 
month,  will  produce  the  required  interest. 

1.  What  is  the  interest  on  $306  for  1  yr.  6  mo.  12  da., 
at  7  per  cent.  ? 

OPERATioisr.  Analysis.     Removing  the  sep- 

1  yr.  6  mo.  12  da.  =  18.4  mo.     aratrix  in  the  given  principal  two 
12  )  S3  OGO  places  to  the  left,  we  have  $3.06, 

— — -^  the  interest  on  the  given  sum  for  1 

$.2o5  year,  at  1  per  cent.     (I.)  Dividing 

^^•^  this  by  12,  we  have  $.255,  the  in- 

^  1020  terest  for  1  month,  at  1  per  cent. 

2040  (IJ-)  Multiplying  this  quotient  by 

255  18.4,  the  time  expressed  in  months 

and  decimals  of  a  month,  (III.)  we 
have  $4,692,  tlie  interest  on  the 
given  sum  for  the  given  time, 
$32.8440,  A71S.  at  1  per  cent.  (IV.)  And  mul- 
tiplying this  product  by  7,  the 
rate  per  cent.,  we  have  $32,844,  the  required  interest. 


4.6920 

7 


184  PEECEKTAGE. 

KuLE.  I.  Remove  the  separatrix  in  the  given  principal 
two  places  to  the  left ;  the  result  is  the  i?iterest  for  1  year 
at  1  per  cejit, 

II.  Divide  this  interest  hy  12  ;  the  result  is  the  interest 
for  1  month,  at  1  per  cent. 

III.  Multiply  this  interest  hy  the  given  time  expressed  in 
months  and  tenths  of  a  month;  the  result  is  the  interest  for 
the  given  time,  at  1  per  cent. 

IV.  Multiply  this  i7iter est  ly  the  given  rate  ;  the  product 
is  the  ifiterest  required. 

Examples  for  Practice. 

2.  "Wliat  is  the  interest  on  $3425  for  3  yr.  8  mo.  15  da., 
at  5  per  cent.  ?  Ans.  $6.35. 

3.  What  is  the  interest  on  1260  for  9  mo.  3  da.,  at  6 
per  cent.  ?  Ans.  $11,826. 

4.  What  is  the  interest  on  $450,  at  6  per  cent.,  for  10 
mo.  18  da.?  Ans.  $23.85. 

5.  What  is  the  interest  on  $372  for  1  yr.  10  mo.  15  da., 
at  7  per  cent?  A71S.  $48,825. 

6.  What  is  the  interest  on  $221.75  for  3  yr.  7  mo.  6  da., 
at  7  per  cent.  ?  Ans.  $55.88. 

7.  What  is  the  interest  on  $267.27  for  6  mo.  24  days,  at 

6  per  cent.  ?  •         Ans.  $9,086. 

8.  What  is  the  interest  on  $365  for  2  mo.  3  days,  at  6 
per  cent.  ?  A7is^$o.83. 

9.  What  is  the  interest  on  $785.10  for  1  yr.  6  mont]  .d 
18  days,  at  5  per  cent.?     *  Ans.  $60,545. 

10.  On  $450  for  3  yr.  7  months,  at  8  per  cent.  ? 

11.  What  is  the  interest  on  $600  for  2  yr.  8  mo.,  a^  7 
per  cent.  ?  Ans.  $112. 

12.  What  is  the  amount  of  $1000  for  9  mo.  15  daj^s,  at 

7  per  cent.  ?  Ans.  $1055.414. 


SIMPLE     II^TEREST.  185 

13.  What  is  the  interest  on  I860  for  6  mo.  6  days,  at  6 
percent?  Ans.  ^26.66. 

14.  What  is  the  interest  on  $137.45  for  8  mo.  27  days, 
at  6  per  cent.  ? 

15.  Find  the  amount  of  $875  for  1  yr.  6  mo.  at  3  per 
cent.  Alls.  $914,375. 

16.  Find  the  amount  of  $350  for  9  mo.,  at  4  per  cent. 

A7ts,  $360.497o 

17.  Find  the  amount  of  $8.50  for  1  yr.  9  mo.  12  da.,  at 
6  per  cent.  Ans.  $9,409. 

18.  Find  the  amount  of  $457  for  1  yr.  4  mo.  24  da.,  at 

6  per  cent.  Ans.  $495,388. 

19.  Find  the  amount  of  $650  for  3  yr.  10  mo.  21  days,  at 

7  per  cent.  A7ts.  $827,049. 

20.  What  is  the  interest  on  $79  for  15  mo.,  at  7  per 
cent.?  Ans.  $6,912. 

21.  Find  the  amount  of  $.86  for  5  mo.,  at  7  per  cent. 

Ans.  $.885. 

22.  What  is  the  interest  on  $78.75  for  1  yr.  9  mo.,  at  4 
percent.?  A7is.  $5.5125. 

23.  What  is  the  interest  on  $1750  for  30  days,  at  9  per 
cent.  ?  Ans.  $13,125. 

24.  Wliat  is  the  interest  on  $3654.25  for  33  days,  at  10 
per  cent.  ?  Ans.  $33,497. 

25.  Find  the  amount  of  $269.50  for  120  days,  at  7  per 
cent.  '  Ans.  $275,788. 

26.  Find  the  amount  of  $1625  for  1  yr.  6  mo.,  at  8  per 
sent.  •  Ans.  $1820. 

For  a  full  treatise  of  Percentage  in  all  its  applications  to  the  business  trans- 
actions of  life,  and  also  for  the  development  and  application  of  those  subjects 
ordinarily  treated  by  arithmetic,  the  pupil  is  referred  to  the  Author's  Progressive 
Practical,  or  Complete  Arithmetics. 


186  PEOMISCUOUS     EXAMPLES. 


PEOMISCUOUS    EXAMPLES, 

1.  Multiply  the  difference  between  876042  and  834260 
by  176.  Ans.  7353632. 

2.  To  47320  add  three  times  the  difference  between 
46270  and  31032,  Ans.  93034. 

3.  From  212462  +  432046,  take  517240—230124. 

4.  Divide  the  sum  of  4802  +  56010  +  20342  by  4  times 
the  difference  between  1200  and  1082. 

A71S.  171III-. 

5.  What  is  the  difference  between  1824624  + 15624  and 
896042  —  12342?  Ans.  956548. 

6.  What  is  the   difference    between    3426  x  284    and 
200104?  ^Ans.  772880. 

7.  Wliat  IS  the  difference  between  3931476-^-556  and 
14x875?  Ans.  5179. 

8.  How  many  times  can  36  be  subtracted  from  11772  ? 

Ans.  327. 
9.  How  many  times  can  8  x  27  be  taken  from  1554768  ? 

10.  Divide  420  x  216  by  43756—42851. 

Ans.  lOO^p 

11.  Multiply  3  times  the  sum  of  4624  +  1036  by  2  times 
the  difference  of  375—296.  Aois.  2682840. 

12.  What  is  the  dift'erence  between  5  times  2.5,  and 
5x25?  A?is.  112.5. 

13.  Multiply  4.05 +  .025 +  1.8  by  2—1.875. 

14.  Divide  5  by  .8  x  .025.  A7is.  250. 

15.  How  many  times  can  1.05  be  taken  from  4.725  ? 

Ans.  4.5  times. 

16.  To  .02  times  32.5  add  5.7  times  16.04—12.0026. 

A71S.  23.66318. 


PROMISCUOUS     EXAMPLES.  187 

17.  What  is  the  difference  between  .675-^.15  and  .23 
X.009?  Ans.  4.49793. 

18.  A  farmer  sold  a  horse  for  $140,  a  coy\^  for  $25,  and 
28  sheep  at  $2.50  a  head  ;  how  much  more  did  he  receive 
for  the  horse  than  for  the  cow  and  slieep?      Ans.  $45. 

19.  A  young  lady  having  $75,  wxnt  out  shopping,  and 
bought  14  yards  of  silk  for  a  dress,  at  $1.50  a  yard,  a 
shawl  for  $15.75,  a  bonnet  for  $8,  a  pair  of  gloves  for 
$1,125,  and  a  pair  of  shoes  for  $1.75  ;  how  much  money 
had  she  remaining  ?  .  Ans.  $27.37 J. 

20.  A  grocer  bouglit  12  firkins  of  butter,  each  contain- 
ing 56  pounds,  at  14  cents  a  pound  ;  he  afterward  sold  5 
firkins,  at  16  cents,  and  7  firkins,  at  18  cents  a  pound; 
what  was  his  whole  gain  f  Ans.  $21. 2S. 

21.  A  miller  sold  256  barrels  of  flour,  at  $6.80  a  barrel, 
which  was  $475.60  more  than  the  wheat  from  which  it 
was  made  cost  him  ;  v/hat  was  the  cost  of  the  wheat  ? 

Ans.  $1265.20. 

22.  An  estate  worth  $25640,  has  demands  againsfc  it  to 
the  amount  of  $9376  ;  after  these  claims  are  paid,  the 
remainder  is  to  be  divided  equally  among  5  individuals ; 
what  will  each  receive  ?  Ans.  $3252.80. 

23.  If  15  tons  of  hay  cost  $311.70,  how  much  will  1  ton 
cost?  A71S.  $20.78. 
-^24.  Paid  $1.24  for  15.5  pounds  of  beef;  what  was  the 
price  per  pound  ?  A?is.  $.08. 

25.  A  farmer  exchanged  21  bushels  of  wheat,  at  $2  a 
bushel,  for  cloth  worth  $3  a  yard  ;  how  many  yards  did 
he  receive  ?  Ans.  14  yards. 

26.  A  man  having  labored  for  a  farmer  1  year,  at  $15  a 
month,  expended  the  year's  wages  for  cows,  at  $18  each  ; 
how  many  cows  did  he  buy  ?  Ans,  10. 


188  PROMISCUOUS     EXAMPLES. 

27.  What  will  be  tlie  cost  of  3  hogsheads  of  sugar,  each 
weighing  15  cwt.,  at  8  cents  a  pound  ?  Ans.  $360.   - 

28.  How  many  bushels  of  wheat,  at  $1.12  a  bushel,  can 
be  bought  for  181.76  ?  A^i^.  73. 

29.  If  140  barrels  of  apples  cost  $329,  what  is  the  cost 
per  barrel?  A7is.  $2.35. 

30.  At  $.825  per  bushel,  how  many  bushels  of  corn  can 
be  bought  for  $264?  Ans.  320. 

31.  If  25  yards  of  cloth  can  be  bought  for  $125.25,  how 
many  yards  can  be  bought  for  $751.50  ?  Atis.  150. 

82.  If  150  bushels  of  wheat  cost  $435,  what  will  311 
bushels  cost  ?  Ans.  $901.90. 

33.  If  250  pounds  of  tea  cost  $135,  what  is  the  price 
per  pound  ?  .  A''a&,  $.54. 

34.  If  3  3  spoons  are  made  from  2  lb.  10  oii.  9  pwt.  of 
silver,  what  is  the  weight  of  each  ? 

Ans,  2  oz.  13  pwt. 

35.  If  a  man  travels  20  mi.  156  rd.  in  a  day,  how  far 
will  he  travel  in  61  days  at  the  same  rate  ? 

Ans.  1249  mi.  236  rd. 

36.  If  I  put  376  gal.  3  qt.  1  pt.  of  cider  into  9  equal 
casks,  how  much  do  I  put  into  each  cask  ? 

37.  If  a  family  use  l-J  pounds  of  tea  in  1  month,  how 
much  would  they  use  in  1  year  ?         Ans.  13^  pounds. 

38.  What  is  the  cost  of  565  pounds  of  butter  at  12^ 
cents  a  pound  ?  Ans.  $70,625. 

39.  At  $4.25  per  bushel  how  much  clover-seed  can  be 
boudit  for  $11.6875  ?  Ans.  2. 75  bushels. 

40.  At  ^  of  a  dollar  a  pound,  what  will  be  the  cost  of 
12  pounds  of  sugar  ?  Ans.  $.75. 

41.  At  I  of  a  dollar  a  yard,  what  will  be  the  cost  of 
40f  yards  of  cloth  ?  Ans,  $15.30. 


PROMISCUOUS     EXAMPLES.  189 

42.  How  many  cubic  yards  of  earth  must  be  thrown 
from  a  cellar  40  ft.  long,  30  ft.  wide,  6  ft.  deep  ;  and  what 
will  be  the  cost  of  the  excavation,  at  12-|-  cents  a  cubic 
yard  ?  Ans.  266f  cubic  yards  ;  $33.33|. 

43.  If  6  pounds  of  cheese  cost  $f,  how  much  will  10 
pounds  cost  ?  Ans.  |1-|-. 

44.  How  much  wheat  at  $1.25  a  bushel,  must  be  given 
for  50  bushels  of  corn  at  $.70  a  bushel  ? 

45.  At  10  cents  a  pint,  what  will  189  gallons  of  molasses 
cost  ?  Ans.  1151.20. 

46.  At  15  cents  a  pound,  what  will  -^  of  a  pound  of 
coffee  cost  ?  Atis.  2f  cents. 

47.  If  3  gallons  of  molasses  cost  Sf ,  how  many  gallons 
can  be  bought  for  $4  ?  A7is.  14|. 

48.  At  $7|-  a  firkin,  how  many  firkins  of  butter  can  be 
bought  for  133  ?  Ans.  4f. 

49.  If  -J  of  a  yard  of  cloth  cost  l-f-^  what  will  one  yard 
cost?  Alts.  82-f-. 

50.  At  $3  a  barrel,  how  many  barrels  of  cider  can  be 
bought  for  |8|  ?  Ans.  2{^  barrels. 

51.  What  part  of  100  pounds  is  16  pounds  ? 

Ans.  ^. 

52.  How  much  wood  in  a  load  10  ft.  long,  3-J-  ft.  wide, 
and  4  ft.  high  ?  A7is.  1  Cd.  12  cu.  ft. 

53.  How  many  tons  of  coal  maybe  bought  for  8346.125 
at  19.75  per  ton  ?  Ajis.  35.5  tons. 

54.  What  is  the  interest  on  $136.80  for  1  yr.  11  mo.,  at 
7  per  cent  ?  •        A7is.  818.354. 

55.  What  Tvill  be  the  cost  of  .6  of  a  gallon  of  wine,  at 
$.65  a  gallon  ?  A?is.  8.39. 

56.  A  owns  4  of  a  flouring  mill,  and  sells  f  of  his  share 
to  B  ;  what  part  of  the  whole  has  he  left  ? 


190  PROMISCUOUS     EXAMPLES. 

57.  If  2  yards  of  clotli  cost  |6f ,  what  will  9  yards  cost  ? 

Ans.  $30|. 

58.  What  will  j-  of  |-  of  a  barrel  of  flour  cost  at  $7^  per 
barrel  ?  A?is.  12 J. 

59.  If  1  acre  of  land  jrield  1  T.  9  cwt.  47  lb.  of  hay, 
how  much  will  18  acres  yield  ? 

60.  A  speculator  bought  1575  barrels  of  potatoes,  and 
upon  opening  them,  he  found  15  per  cent,  of  them  spoiled ; 
how  many  barrels  did  he  lose?  Ans.  236.25. 

61.  How  many  steps  of  30  inches  each,  must  a  person 
take  in  walking  10  miles  ?  A?is.  21120. 

62.  A  man  bought  12  bushels  of  chestnuts,  at  $4.50  a 
bushel,  and  sold  them  at  12  cents  a  pint ;  what  was  his 
whole  gain?  Ans.  $38.16. 

63.  What  is  the  interest  on  $300,  for  10  mo.  21  days, 
at  6  per  cent.?  Ans.  $16.05. 

64.  An  agent  in  Chicago,  purchased  5450  bushels  of 
wheat,  at  $.82  a  bushel ;  what  was  his  commission  at  2  per 
cent,  on  the  purchase  money  ?  Ans.  $89.38. 

65.  A  vessel  loaded  with  4500  bushels  of  corn,  was 
overtaken  by  a  storm  at  sea,  and  it  was  found  necessary 
to  throw  overboard  25  per  cent,  of  her  cargo  ;  what  was 
the  whole  loss,  at  60  cents  a  bushel  ?  Ajis.  $675. 

66.  A  grocer  bought  2  hogsheads  of  molasses,  at  37-|- 
cents  a  gallon,  and  sold  it  at  20  per  cent,  advance  on  the 
cost ;  what  was  his  whole  gain?  Ans.  $9.45. 

67.  If  f  of  an  acre  of  land  is  worth  $60,  what  is  the 
value  of  1  acre  ?  Ans.  $84. 

68.  If  1-J-  bushels  of  wheat  sow  an  acre  of  land,  how 
many  acres  will  12  bushels  sow?  A7is.  9  acres. 

69.  If  a  farm  is  worth  $3840,  what  is  |  of  it  worth  ? 

Ans.  $2400. 


PKOMISCUOUS     EXAMPLES.  191 

70.  If  17  kegs  of  nails  weigh  27  cwt.  3  qrs.  23  lbs.  3  oz., 
long  ton  weight,  how  much  will  1  keg  weigh  ? 

71.  If  a  bushel  of  apples  cost  f  of  a  dollar,  how  many 
may  be  bought  for  f  of  a  dollar  ? 

72.  Divide  |  of  f  by  f  of  J.  A^is.  f 

73.  What  is  the  amount  of  $620  for  4  yr.  3  mo.,  at  6 
per  cent.  ?  Ans.  $778.10. 

74.  What  is  the  brokerage  on  $5462,  at  4  per  cent.  ? 

75.  How  many  pounds  of  butter  at  13  J  cents  a  pound, 
must  be  given  for  1230  pounds  of  sugar  at  8  cents  a 
IDOund  ?  Ajis.  72 8f  pounds. 

76.  Divide  168  bu.  1  pk.  6  qt.  of  corn  equally  among  35 
persons.  Ans.  4  bu.  3  pk.  2  qt. 

77.  What  will  be  the  cost  of  lathing  and  plastering 
overhead,  a  room  36  feet  long  and  27  feet  wide,  at  28 
cents  a  square  vard?  A71S.  $30.24. 

78.  How  much  land  at  $2.50  an  acre,  must  be  given  in 
exchange  for  360  acres,  at  $3.75  an  acre? 

79.  What  is  the  amount  of  $564.58,  for  3  yr.  5  mo.  12 
da.,  at  6  per  cent.?  A71S.  $681,448. 

80.  How  much  sugar  at  9  cents  a  pound,  should  be 
given  for  6^  cwt.  of  tobacco,  at  14  cents  a  pound  ? 

81.  How  many  times  may  a  jug  which  holds  -J  of  a 
gallon,  be  filled  from  a  cask  containing  128  gallons  ? 

82.  A  man  ha^dng  $25000,  invested  30  per  cent,  of  it  in 
bonds  and  mortgages,  45  per  cent,  of  it  in  bank  stocks, 
and  the  remainder  in  railroad  stock  ;  how  much  did  he 
invest  in  railroad  stock  ? 

A71S.  $6250. 

83.  How  many  times  can  a  box  holding  4  bu.  3  pk.  2  qt., 
be  filled  from  336  bu.  3  pk.  4  qt.  ? 

Ans.  70. 


192  PEOMISCUOUS     EXAMPLES. 

84.  How  many  cords  of  wood  in  17  piles,  each  11  feet 
long,  4  feet  wide,  and  6  feet  high  ? 

85.  If  the  price  of  1  acre  of  land  is  I32f ,  what  is  the 
value  of  I  of  an  acre  ?  Ans.  $28|J. 

86.  What  number  of  times  will  a  wheel  14  ft.  10  in.  in 
circumference,  turn  round  in  traveling  11  mi.  255  rd. 
12  ft.  6  in.  ?  Ans.  4200. 

87.  A  man  bought  a  farm  of  136  acres,  at  194  an  acre ; 
he  paid  1475  for  fencing  and  the  improvements,  and  then 
sold  it  for  14  per  cent,  advance  on  the  whole  cost ;  what 
was  his  whole  gain?  Ans.  11856.26. 

88.  If  36.48  yards  of  cloth  cost  $54.72,  what  will  14.25 
yards  cost?  Ans.  121.375. 

89.  If  $13,342  will  pay  for  17.5  bushels  of  barley,  how 
many  bushels  can  be  bought  for  $76.24  ? 

Ans.  100  bushels. 

90.  A  lady  having  $40. 50,  spent  40  per  cent,  of  it  for 
dry  goods  ;  what  had  she  left?  Ans.  $24.30. 

91.  A  gentleman  bought  a  house  and  lot  for  $6425  ;  in 
the  course  of  five  years  it  increased  in  value  110  joer  cent. ; 
what  was  the  property  then  worth  ?       Ans.  $13492.50. 

92.  What  will  a  broker  charge  to  change  $560  uncur- 
rent  money  for  current  money,  at  3  per  cent.  ? 

Ans.  $16.80. 

93.  If  4  hogsheads  of  wine  cost  $181.44,  what  is  the 
cost  of  1  pint  ?  Ans.  9  cents. 

94.  What  will  5  casks  of  rice  cost,  each  weighing  165 
pounds,  at  5^  cents  a  pound?  Ans.  $45.37-|. 


METRIC     SYSTEM.  193 


METEIO    SYSTEM. 

198.  The  Metric  System  of  weights  and  measures  is 
based  upon  the  decimal  scale. 

199.  The  Meter  is  the  dase  of  the  system,  and  is  the 
one  ten-millio7itli  part  of  the  distance  on  the  earth's  sur- 
face from  the  equator  to  either  pole,  or  39.37079  inches. 

200.  From  the  Meter  are  made  the  Ai^e  (air),  the 
Stere  (stair),  the  Liter  (leeter),  and  the  Gram;  these 
constitute  the  j^rimary  or  prijicipal  units  of  the  system 
from  which  all  the  others  are  derived. 

201.  The  Multiple  Units,  or  higher  denominations, 
are  named  by  prefixing  to  the  name  of  the  primcu^y  units 
the  Greek  numerals,  DeJca  (10),  Hecto  (100),  Kilo  (1000), 
and  Myra  (10000). 

203.  The  Sub-multiple  Units,  or  lower  denomina- 
tions, are  named  by  prefixing  to  the  names  of  the  pri- 
mary units  the  Latin  numerals,  Deci  (yV)>  Centi  (y^i)-), 
Mille  (toVo). 

Hence,  it  is  apparent  from  the  name  of  a  unit,  whether 
it  is  greater  or  less  than  the  standard  unit,  and  also  Jioiv 
many  times. 

The  following  are  the  metric  measures  authorized  by 
Congress  in  1866,  with  their  equivalents : 


194 


METRIC     SYSTEM. 


MEASUEES    OF    EXTENSION. 

303.  The  Meter  is  the  unit  of  length,  and  is  equal  to 

39.37  in.,  nearly. 

Table. 


Metric  Denominations. 


10  MilUmeters, 

mm.     =  1 

10  Centimeters, 

cm.      =  1 

10  Decimeters, 

dm.      =  1 

10  Meters, 

if.       =  1 

10  Dekameters, 

Dm.     =  1 

10  Hectometers, 

Rm.    =  1 

10  Kilometers 

Km.    =  1 

U.  S.  Value. 
.03937079  in. 
.3937079  in. 
3.937079  in. 
39.37079  in. 
32.808992  ft. 
19.927817  rd. 
.6213824  mi. 
6.213824  mi. 


Millimeter 
Centimeter 
Decimeter 
Meter 
Dekameter 
Hectometer 
Kilometer 
1  Myriameter  (Mm.) 

The  Meter,  like  our  yard,  is  used  in  measuring  cloths 
and  short  distances. 

-    The  Kilometer  is  commonly  used  for  measuring  long 
distances,  and  is  about  |-  of  a  common  mile. 

204.  The  Are  is  the  unit  of  land  measure,  and  is  a 
square  whose  side  is  10  meters,  equal  to  a  square  deka- 
meter, or  119.6  square  yards. 

Table. 

1  Centiare,  ca.     -  (1  Sq.  Meter)        =       1.196034  sq.  yd 
100  Centiares,  "      =  1  Are  =  119.6034  sq.  yd. 

100  Ares         A.     =  1  Hectare  (Jla.)  —      2.47114  acres. 

205.'  The  Stere  is  the  unit  of  wood  or  solid  measure, 

and  is  equal  to  a  cuMc  meter,  or  .2759  cord. 

Table. 

1  Decistere  =     3.531  +  cu.  ft. 

10  Decisteres,  dst  =  1  Stere  =  35.316+  cu.  ft. 

10  Steres,       8t.    =  1  Dekastere  (BSt.)   =  13.079+  cu.  yd. 

The  Square  Meter  is  the  unit  for  measuring  ordinary 
surfaces  ;  as  flooring,  ceilings,  etc. 

The  Cubic  Meter  is  the  U7iit  for  measuring  ordinary 
solids  ;  as  excayations,  embankments,  etc. 


METEIC     SYSTEM. 


195 


MEASUEES    OF    CAPACITY. 

206.  The  Liter  is  the  unit  of  capacity,  both  of 
Liquid  and  of  Dry  Measures,  and  is  a  vessel  whose 
volume  is  equal  to  a  cube  whose  edge  is  one-tenth  of  a 
meter,  equal  to  1.05673  qt.  Liquid  Measure,  and  .9081  qt. 
Dry  Measure. 

Table. 


10  Milliliters, 

ml.     . 

.    .     =  1  Centiliter. 

10  Centiliters, 

d.      . 

.    .     =  1  Deciliter. 

10  Deciliters, 

dl.     .    . 

.    =  1  Liter. 

10  Liters, 

L.      . 

.     .     =  1  Dekaliter. 

10  Dekaliters, 

Bl.    . 

.     .     =  1  Hectoliter. 

10  Hectoliters, 

m.   . 

.     .     =  1  Kiloliter,  or  Stere 

10  Kiloliters, 

Kl.     . 

.     .     =  1  Myrialiter  {Ml) 

The  Hectoliter  is  the  unit  in  measuring  liquids,  grain, 
fruit,  and  roots  in  large  quantities. 


Equivalents  in  United  States  Measures. 


Metric  Denom.  Cubic  Measure. 

1  Myrialiter  =  10  cubic  meters 
1  Kiloliter  =  1  cubic  meter 
1  Hectoliter  = 
1  Dekaliter  = 
1  Liter  = 
1  Deciliter  = 
1  Centiliter  = 
1  Mniiliter    = 


-^^  cubic  meter  = 
10  cu.  decimeters   ■= 

1  cu.  decimeter  = 
-^  cu.  decimeter  = 
10  cu.  centimeters  = 

1  cu.  centimeter   = 


Dry  Measure. 
13.08+  cu.yd. 
28.372+  bu. 
2.8372+  bu. 
9.08  quarts 
.908  quart 
6.1022  cu.  in. 
.6102  cu.  in. 
.061  cu.  in. 


Wine  Measure. 
=  2641 .4  + gal. 
=  264.17  gal. 
:  26.417  gal. 
:  2.6417  gal. 
=  1.0567  qt. 
-.  .845  gill. 
-.  .338  fluid  oz. 
:  .27  fluid  dr. 


196 


METRIC     SYSTEM. 


MEASURES    OF    WEIGHT. 

207.  The  Gram  is  the  imit  of  iveiglit,  and  equal  to 
the  weight  of  a  cube  of  distilled  water,  the  edge  of  which 
is  one  hundredth  of  a  meter,  equal  to  15.432  Troy  grains. 

Table. 


IT.  S.  Valae 

10  Milligrams 

mg. 

T=zl  Centigram 

=:         .15432+  gr. 

Troy. 

10  Centigrams, 

eg. 

=  1  Decigram 

=       1.54324+    " 

<  < 

10  Decigrams, 

dg. 

=1  Gram 

=     15.43248+    '' 

( ( 

10  Grams 

0. 

=1  Dekagram 

=         .35273+  oz. 

Avoir 

10  Dekagrams, 

Dg. 

-—\  Hectogram 

=      3.52739+   '' 

(  s 

10  Hectograms, 

Hg. 

j  Kilogram  ) 
~^  1  or,  Kilo  \ 

=      2.20462+  lb. 

(( 

10  Kilograms, 

Kg. 

=  \  Myriagram 

=     22.04621+   " 

(C 

10  Myriagrams,orJ^. 
100  Kilograms 

\ 

=  1  Quintal 

=  220.46212+   " 

(I 

10  Quintals,  or 

\ 

.  {  Tonneau,  ) 
\   or  Ton   \ 

=2204.62125      " 

<( 

1000  Kilos 

The  Kilogrmn,  or  Kilo,  is  the  unit  of  common  weight 
in  trade  and  is  a  trifle  less  than  2-J-  lb.  Avoirdupois. 

The  Tonneau  is  used  for  weighing  very  heavy  articles, 
and  is  about  204  lb.  more  than  a  common  ton. 

208.  Units  of  the  Common  System  may  be  readily 

changed  to  units  of  the  MetiHc  System  by  the  aid  of  the 

following 

Table. 


1  Inch        =  2.54  Centimeters. 
1  Foot         =  30.48  Centimeters. 
1  Yard        =  .9144  Meter. 
1  Rod         =  5.029  Meters. 
1  Mile         =  l.()093  Kilometers. 
1  Sq.  inch  =  6.4528  Sq.  Centimet. 
1  Sq.  foot  =  929  Sq.  Centimeters. 
1  Sq.  yard  =  .8361  Sq.  Meter. 
1  Sq.  rod    =  25.29  Centiares. 
1  Acre        =  40.47  Ares. 
1  Sq.  mile  =  250  Hectares. 


Cu.  inch  : 
Cu.  foot  : 
Cu.  yard  : 
Cord 

Fl.  ounce: 
Gallon 
Bushel 
Troy  gr. 

1  Troy  lb.   : 

1  Av.  lb. 

1  Ton 


16.39  Cu.  Centimet. 
28320  Cu.  Centimet. 
.7646  Cu.  Meter. 
3.635  Steres. 
2.958  Centiliters. 
3.786  Liters. 
.3524  Hectoliter 
64.8  Milligrams 
.373  Kilo. 
.4586  Kilo. 
.907  Tonneau. 


MISCELLAI^TEOUS.  197 


MISCELLAINEOUS  TABLES. 

209.  The  old  French  Linear,  and  Land  Measure, 

is  still  used  to  some  extent  in  Louisiana,  and  in  other 
French  settlements  in  the  United  States. 


Table. 

12  Lines     =  1  Inch.  6  Feet      =  1  Toise. 

13  Inches  =  1  Foot.  32  Toises  —  1  Arpent. 

900  Square  Toises  =  1  Square  Arpent. 

The  French  Foot  equals  12.8  inches,  American,  nearly. 
The  Arpent  is  the  old  French  name  for  Acre,  and  contains  nearly 
f  of  an  English  acre. 

In  Texas,  New  Mexico,  and  in  other  Spanish  settle- 
ments of  the  United  States,  the  following  denominations 
are  still  used  : 


Table. 

1000000  Square  Varas  =  1  Labor     =    177.136  Acres  (American). 
25  Labors  =  1  League  =  4428.4      Acres  " 

The  Spanish  Foot  =  11.11+  in.  (Am.)  ;   1  Vara  =  331  in.  (Am.); 
108  Varas  =  100  Yards,  and  1900.8  Varas  =  1  Mile. 

Other  Denominations  in  Use. 

* 

5000         Varas  Square  =        1  Square  League. 
1000         Varas  Square  =        1  Labor,  or  4^  League. 
5645.376  Square  Varas  -  4840  Square  Yards  =      1  Acre. 
23.76    Square  Varas  =        1  Square  Chain  =    y^^  Acre. 
1900.8      Varas  Square  =        1  Section  =  640  Acres. 


198 


MISCELLANEOUS. 


310.  The  following  table  will  assist  farmers  in  making 
an  accurate  estimate  of  the  amount  of  land  in  different 
fields  under  cultivation  : 

Table. 


10  rods 

X 

16  rods  = 

A. 

220 

feet    X 

198 

feet 

=  1   A 

8     " 

X 

20     "      = 

no 

"          X 

369 

'~'~      J_ 

5     " 

X 

32     "     = 

60 

'*          X 

726 

;; —      J_ 

4     " 

X 

40     "     = 

120 

"          X 

363 

;           J_ 

5  yds. 

X 

968  yds.   = 

200 

"          X 

108.9 

=  1    " 

10     " 

X 

484     "      = 

100 

"          X 

145.2 

r=    1     " 

26     " 

X 

242     "      = 

100 

"          X 

108.9 

X 

40     " 

X 

121     "     = 

311.  The  following  table  will  often  be  found  conven- 
ient, taking  inside  dimensions  : 

A  box  24  in.  x  24  in.   x  14.7  will  contain  a  harrel  of  31|  gallons. 

A  box  15  in.  x  14  in.  x  11  in.  will  contain  10  gallons. 

A  box  8^  in.  x  7  in.  x  4  in.  will  contain  a  gallon. 

A  box  4  in.   x  4  in.  x  3.6  in.  will  contain  a  quart. 

A  box  24  in.  x  28  in.   x  16  in.  will  contain  5  bushels. 

A  box  16  in.   x  12  in.  x  11.2  in.  will  contain  a  bushel. 

A  box  12  in.   x  11.2  in.  x  8  in.  will  contain  a  half-bushel. 

A  box  7  in.  x  6.4  in.   x  12  in.  will  contain  a  'peek. 

A  box  8. 4  in.  x  8  in.  x  4  in.  will  contain  a  half -peck  or  4  dry  quarts. 

A  box  6  in.  by  5|  in.,  and  4  in.  deep,  will  contain  a  half  gallon. 

A  box  4  in.  by  4  in.,  and  2y\f  in.  deep,  will  contain  a  pint. 

213.  Nails  are  put  up  100  pounds  to  the  keg. 


^ 

^ 

. 

^ 

, 

Si    re' 

rjl^ 

■^    Xf2 

tfj^ 

-G  a 

:fu^ 

Size. 

Size. 

Size. 

1- 

•1— < 

S  03 

725 

16 

Qd  casing. 

2 

3c?  fine  blued. 

30(i  com.  blued. 

210 

8fZ  com. 

^ 

400 

40fZ     " 

5 

14 

M      " 

n 

134 

Ad    " 

u 

300 

^M     " 

^ 

11 

lOd      " 

78 

Qd    " 

2' 

150 

md     " 

6 

8 

Qd  finishing 

2 

317 

M    « 

21. 

85 

M  fence. 

2 

80 

M     " 

2i 

208 

IM  " 

3^ 

60 

M      " 

2i 

50 

IQd     '•' 

3" 

126 

l%d  " 

3:1 

50 

lOi       '• 

3 

30 

Qd  clinching 

2 

118 

IM  " 

3^ 

40 

IM      " 

81 

27 

M      " 

<Z\ 

80 

20(Z  " 

4 

20 

16^       " 

^ 

20 

\M      " 

3 

45 

5  lbs.  of  U  or  3  J  lbs.  of  M  will  put  on  1,000  shingles. 
5f  lbs.  of  M  fine  will  put  on  1,000  lath. 


MISCELLANEOUS. 


199 


Eailroad  Freight. 

213.  When  convenient  to  weigli  them,  all  goods  are 
billed  at  actual  loeight ;  but  ordinarily,  the  articles  named 
below  are  billed,  at  the  rates  given  in  the  following 

Table. 


Ale  or  Beer, 

820  lbs. 

per  bbl. 

Highwmes, 

350  lb£ 

5.  per  bbl. 

Apples,  green, 

150  " 

(t 

Lime, 

200   " 

ti 

Beef, 

320   " 

a 

Nails, 

108  " 

per  keg. 

Barley, 

48   " 

per  bu. 

Oil, 

400   " 

per  bbl. 

Beans, 

60   " 

« 

Oats, 

82  " 

per  bu. 

Cider, 

850  " 

per  bbl. 

Pork, 

320  " 

per  bbl. 

Corn  Meal, 

220  " 

(< 

Potatoes,  com'r 

1,150   " 

(( 

Corn,  shelled, 

56  " 

per  bu. 

Salt,  fine, 

300  " 

(( 

Corn  in  ear. 

70  " 

a 

' '     coarse. 

350   " 

<i 

Clover  Seed, 

60   " 

<( 

"     in  sacks. 

200  " 

per  sack. 

Eggs, 

200  " 

per  bbl. 

Wheat, 

60   " 

per  bu. 

Fish, 

800  " 

(< 

Whiskey, 

350   " 

per  bbl. 

Flour, 

200   " 

i( 

2000  pounds  are  reckoned  1  Ton. 

Generally  from  18000  to  20000  pounds  is  considered  a  car  load. 


214.  Lumber  and  some  other  articles  are  estimated  as 
follows  : 


Weight. 
Pine,  Hemlock,  and  Poplae,  thoroughly 

seasoned,  per  thousand  feet 3000 

Black  Walnut,  Ash,  Maple,  and  Cheery, 

per  thousand  feet 4000 

Pine,  Hemlock,  and  Poplar,  green,  per  M .  4000 
Black  Walnut,  Ash,  Maple,  and  Cherry, 

green,  per  M 4500 

Oak,  Hickory,  and  Elm,  dry,  per  M 4000 

Oak,  Hickory,  and  Elm,  green,  per  M 5000 

Shingles,  green,  per  thousand 375 

Lath,  per  thousand. 500 

Brick,  common,  per  car  load 4  lbs  each 

Coal,  per  car  load 

Stone,  undressed,  per  cubic  yard 4000 


Amount  for 
car  load, 

6500 


5000 

5000 

4000 

5000 

4000 

35  M. 

40  M. 

5000 

250  bu. 

5  cu.  yd. 

Table  for  Investors. 

215.  Thefolloimig  Table  shows  the  rate  per  cent,  of  Annual  Income 
from  Bonds  hearing  5,  6,  7,  or  8  per  cent,  interest,  and  costing 
from  40  to  125. 


Purchase 
Price. 

5%. 

6%. 

7%. 

8%. 

Purchase 
Price. 

5/^. 

Qfo. 

7.22 

Ifo. 
8.43 

8%. 

40 

12.50 

15.00 

17.50 

20.00 

83 

6.02 

9.63 

41 

12.20 

14.64 

17.08 

19.52 

84 

5.95 

7.14 

8.33 

9.52 

43 

11.90 

14.28 

16.66 

19.04 

85 

5.88 

7.05 

8.23 

9.41 

48 

11.63 

13.95 

16.28 

18.61 

86 

5.81 

6.97 

8.13 

9.30 

44 

11.36 

13.63 

15.90 

18.18 

87 

5.74 

6.89 

8.04 

919 

45 

11.11 

13.33 

15.56 

17.78 

88 

5.68 

6.81 

7.94 

9.09 

46 

10.86 

13.04 

15.21 

17.39 

89 

5.61 

6.74 

7.86 

8.98 

47 

10.63 

12.77 

14.90 

17.02 

90 

5.55 

6.66 

7.77 

8.88 

48 

10.41 

12.50 

14.53 

16.66 

91 

5.49 

6.59 

7.69 

8.79 

49 

10.20 

12.25 

14.29 

16.33 

92 

5.43 

6.52 

7.60 

8.69 

50 

10.00 

12.00 

14.00 

16.00 

93 

5.37 

6.45- 

7.52 

8.60 

51 

9.80 

11.76 

13.73 

15.68 

94 

5.31 

6.38 

7.44 

8.51 

52 

9.61 

11.53 

13.46 

15.38 

95 

5.26 

6.31 

7.36 

8.43 

53 

9.43 

11.32 

13.20 

15.09 

96 

5.30 

6.35 

7.29 

8.33 

54 

9.25 

11.11 

12.96 

15.81 

97 

5.15 

6.18 

7.21 

8.24 

55 

9.09 

10.90 

12.72 

14.54 

98 

5.10 

6.13 

7.14 

8.16 

56 

..  8.93 

10.70 

12.50 

14.28 

99 

5.05 

6.06 

7.07 

8.08 

57 

8.77 

10.52 

12.27 

14.03 

100 

5.00 

6.00 

7.00 

8.00 

58 

8.ij^ 

10.34 

12.06 

13.79 

101 

4.95 

5.94 

6.93 

7.92 

59 

8.'47 

10.16 

11.86 

13.55 

102 

4.90 

5.88 

6.80 

7.84 

60 

8.33 

10.00 

11.66 

13  33 

103 

•1.85 

5.83 

6.79 

7.76 

61 

8.19 

9.83 

11.47 

13.11 

104 

4.80 

5.76 

6.72 

7.69 

62 

8.06 

9.07 

11.29 

12.90 

105 

4.76 

5.71 

6.66 

7.61 

63 

7.93 

9.52 

11.11 

12.69 

106 

4.71 

5.66 

6.60 

7.54 

64 

7.81 

9.37 

10.93 

12.50 

107 

4.67 

5.60 

6.54 

7.47 

65 

7.69 

9.23 

10.76 

12.30 

108 

4.63 

5.55 

6.48 

7.40 

66 

7.57 

9.09 

10.60 

12.12 

109 

4.58 

5.50 

6.42 

7.33 

67 

7.46 

8.95 

10.44 

11.94 

110 

4.54 

5.45 

6.36 

7.27 

68 

7.35 

8.82 

10.29 

11.76 

111 

4.50 

5.40 

6.30 

7.20 

69 

7.24 

8.69 

10.14 

11.50 

112 

4.46 

5.35 

6.25 

7.14 

70 

7.14 

8.57 

10.00 

11.43 

113 

4.42 

5.30 

6.19 

7.07 

71 

7.04 

8.45 

9.85 

11.26 

114 

4.38 

5.36 

6.14 

7.01 

72 

6.94 

8.33 

9.73 

11.11 

115 

4.35 

5.31 

6.08 

6.95 

73 

6.84 

8.21 

9.58 

10.95 

116 

4.31 

5.17 

6.03 

6.89 

74 

6.75 

8.10 

9.45 

10.80 

117 

427 

5.12 

5.98 

6.83 

75 

6.66 

8.00 

9.33 

10.66  ! 

118 

4.23 

5.08 

5.93 

6.77 

76 

6.57 

7.89 

9.21 

10.52 

119 

4.20 

5.04 

5.88 

6.73 

77 

6.49 

7.79 

9.00 

10.38 

120 

4.16 

5.00 

5.83 

6.66 

78 

6.41 

7.69 

8.97 

10.25 

121 

4.13 

4.95 

5.78 

6.61 

79 

6.32 

7.59 

8.86 

10.12 

122 

4.09 

4.91 

5.73 

6.55 

80 

6.25 

7.50 

8.75 

10.00 

123 

4.06 

4.87 

5.69 

6.50 

81 

6.17 

7.40 

8.64 

9.87 

124 

4.03 

483 

5.65 

6.45 

82 

6.09 

7.31 

8.53 

9.75 

125 

4.00 

4.80 

5^ 

6.40 

et^  ;  ^  —  ./Cjiaa 


/ 


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/ 


.'^r/Al „      ,.     ,  .   ,  .... 


/  /?. 


v^-iyni 


!\ 


f\ 


6^ 


V 


P 


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r^ 


^^  As  Familiar  to  the  Schools  of  the   United  States  as 

Household  JVords.'* 


*^. 


"t 

» 


Robinson's  Progressive  Course 


OF 


MATHEMATICS. 


-►♦•^ 


ROBINSON'S    PROGRESSIvl:    COURSE 
OF    MATHEMATICS,  being  the  most  complete  and  scientific 
course  of  Mathematical  Text -books  published,  is  more  extensively  used  in 
the  Schools  and  Educational  Institutions  of  the  United  States  than  any 
'  ;;r  compel.ing  series. 

,>-  i  In  its  preparation  two  objects  were  kept  constantly  in  view  :  First,  to 

'^  Jurnish  a  full  and  complete  Series  of  Text-Books,  which  should  be  sufficient 

to  give  the  pupil  a  thorough  and  j>ractical  tusiness  education  :  Second,  to 

secure  that  intellectual  culture  without  which  the  mere  acquisition  of  book 

knowledge  is  almost  worthless. 

All  the  improvements  of  the  best  modern  Text-Books,  as  well  as  many 
■neiu  and  original  methods,  and  practical  operations  not  found  in  other  sim- 
ilar  works,  have  been  incorporated  into  tJiese  books,  and  no  labor  or  ex- 
pense has  been  spared  to  give  to  the  public  a  clear,  scientific,  comprehen- 
sive and  complete  system,  not  incumbered  with  unnecessary  theories,  but 
combining  and  systeraa-tizing  real  iviprovements  of  a  practical  and  useful 
kind. 


•  »> 


Robins 07ts  Shorter  Course, 

In  order  to  meet  a  demand  from  many  quarters  for  a  Series  of  Arith- 
metics, few  in  number  and  comprehensive  in  character,  we  have  published 
the  above  course,  in  TWO  books,  in  which  Oral  and  Written  Arithmetic  is 
combined.  These  books  have  met  with  very  great  popularity,  having  been 
ntroduced  into  several  of  the  largest  cities  in  the  United  States,  'J'uey  are 
unusually  handsome  in  get-up,  and  are  substantially  bound  in  cloth. 

*#*  Descriptive  Circulars  and  Price  Lists  will  be  forwarded  to  Teach~ 
t.'S  and  Educationists  on  application.  The  most  liberal  terms  will  be  made 
for  introductiony  exchange  and  examination. 

IYISON5  BLAKEMAN,  TAYLOR  &  CO., 

PUBLISHERS, 

'^  #  New  York  and  Chicago. 


•;*t> 


